IGUYEN HUYOOAN (Chu bien) PHAM TH! BACH NGOC - DOAN QUYNH OANG HUNG THANG - LLTU XUAN flNH
H i SO
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VlfT NAM
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NGUYfiN HUY DOAN {Chu bien) PHAM THI BACH NGOC - DOAN QUYNH - DANG HUNG THANG - LUU X U A N
BAI T A P
DAI s o NANGCAO (Tdi ban Idn thirndm)
NHA XUATbooktoan.com BAN GIAO DUC VI^T NAM
TINH
Ban quy^n thuoc Nha xu^'t ban Giao due Vidt Nam 01-2011/CXB/850-1235/GD
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Ma s6 : NB003T1
Ld I NOI
DAU.
Ki til nam hoc 2006 2007, ng^h Gi^o due bat ddu thuc hi^n giang day theo chucrng tiinh va sach gi^o khoa mdi Icfp 10. Di khm v6i viec d6i mod chirong trinh va sach giao khoa la ddi mdi v^ phiicfng phap day hoc va d(5i mdi c6ng tdc kilm tra danh gia k6t qua hoc tap cua hoc sinh. Di^u 66 phai duac th^ hi6n khong nhCrng trong sach giao khoa, sach giao vien mh. con trong ca sach bai tap - mOt tiii li6u kh6ng the thieu d6i vdfi giao viSn vk hoc sinh. Cu6'n Bai tap Dai so JO ndng cao nay diroc bi6n soan theo tinh thdn do. Bdii tdp Dai so 10 ndng cao g6m cac bai tap ducfc chon loc va sap x6'p m6t each h6 th6'ng, bam sat tiing chu d6 kid'n thiic trong sach giao khoa, nh^m giiip cac em hoc sinh sir dung song song vdri s^ch giao khoa, vira Cling c6 ki6'n thiic dang hoc, viJta nAng cao ki nang giai toAn. Titong tu nhu sach gi^o khoa Dai sd' 10 ndng cao, noi dung cua sach n^y g6m sau chirong : Chucrng I. Menh d^ - Tap hop Chuong II. Ham sd bac nha't va bac hai Chircmg HI. Phuong trinh v& he phuomg trinh Chucfng IV. B^t dang thirc vk bait phuong trinh Chucrng V. Th6'ng ke Chucfng VI. G6c lucmg giac va c6ng thiic lucmg giac. M6i chuong d^u ducrc md d^u bang ph^ "Nhihig kien thiJfc can nhd" P h ^ n&y t6m tat lai nhutig kiS'n thiic quan trong cua chuofng. Hoc sinh doc "Nhung kien thitc can nh&" d^ tim toi nhfing ki6'n thiic duoc van dung trong qua trinh giai bai tap. Sau khi hoc xong m6i chuong, cac em n6n tr6 lai phdn nay de' 6n tap vk ghi nhd nhirng kie'n thiic do. booktoan.com
Tie'p theo la p h ^ "De bai" va sau do la p h ^ "Dap sd'- Huong dan Ldi giai". Cac bai tap trong phdn "De bai" duoc sap xep theo dung trinh tu cac bai hoc trong sach gido khoa. Do do hoc sinh c6 thd de dang tu lua chpn bai tap d^ lam th6m sau m6i bai hoc. Ben canh cac bai tap bam sat y^u cdu cua sach giao khoa, sach con bo sung m6t s6' bai tap vdi yeu cdu cao ban, giup hoc sinh bu6c ddu tiep can vdri nhiJng dang toan chu^n bi thi vao Dai hoc. Ngoai ra, cu6'i m6i chuong d6u c6 cdc bai tap trac nghi6m khach quan nham giup hoc sinh lam quen vol phuong phap kiem tra danh gia mdi nay. CAn chii y rang m6i cau hoi trac nghi^m khach quan, hoc sinh chi duoc Jam trong thcfi gian he't sire ban ch^ (chang ban, tir 1 de'n 2 phut). Sau khi giai bai tap, hoc sinh c6 the' tu minh ki^m tra lai ke't qua bang each d6'i chieu vdi ph^n "Ddp s6'- Hudng din - Left giai" (ngay sau phdn "De bai" cua m6i chuong). Trong phSn nay, cac tac gia chi chpn loc va nSu led giai d^y dit ciia m6t s6' it bai, eon lai p h ^ 16n cac bai d^u chi cho ddp s6' hoac dap s6' c6 \ahca. theo gpi y khi c^n thie't. Chu y rang cac hu6ng giai duoc neu trong "Huang ddn'\ tham chi trong cdc bai giai chi ti^t cung CO thI chua phai la hudng giai t6't nhSt. Cac tac gia n h ^ manh di^u nay vdi mong mu6'n : chinh hoc sinh se la nhftng ngudi dua ra nhftng Icri giai hay hon, sdng tao hon. Mac du cac tac gia da nit kinh nghidm tijt sach thf di^m va da c6' gang dl c6 duoc ban thao tO't nha't, nhung chae chin sach khdng tranh khoi con nhi^u thie'u sot. Cac tac gia ra't mong nhan dupe gop y cua ban doc g&i xa, nha't la ciia giao vien va cac em hoc sinh - nhOng ngucri true tie'p sijr dung sach. Cu6'i cung, cac tac gia to long bie't on.d^n H6i d6ng T h ^ dinh ciia BO Giao due - Dao tao da gop nhilu y kie'n quy bau, ddn Ban bidn tap sach Toan Tin, C6ng ty c6 p h ^ Dich vu xuSit ban Giao due Ha N6i Nha xu^t ban Giao due Viet Nam da giup dd, hpp tac tich cue va c6 hieu qua trong qua trinh bien soan cu6n Bai tap Dai sd'lO ndng cao nay. CAC TAC GlA
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Q^huan^I MENH DE - TAP HOP
A. N H O N G KIEN THQC CAN NHO Menh de • Menh d^ logic (gpi tat la menh d^) la m6t eau khang dinh dung hoac mdt eau khang dinh sai. M6t menh d^ khOng the' viifa dung viita sai. • Menh dd "Kh6ng ph^i F\ ki hieu l a ? , dupe gpi la menh de phu dinh cua P. Menh dd P dung ne'u P sai va P sai neu P dung. • Menh dd "Ne'u P thi Q", ki hieu la/^ => Q, dupe gpi la menh dd keo theo. Menh dd k^o theo chi sai khi P diing, Q sai, • Menh dd "P ne'u va ehi ne'u Q\ ki hieu l a f o g , dupe gpi la menh dd tuong duong. Menh dd nay dung khi va ehi khi P, Q ciing dung hoac cung sai. Phu dinh cua menh dd " VJC G X, P{x)" la menh dd " 3x e X, P{x) • • Phii dinh cua menh dd " 3x & X, P{x)" la menh dd " Vx e X, P{x)"
Tap.hdp • Tap A dupe gpi la tap con ciia tap B, ki hieu la A c 5, ne'u mpi phan tijf cua A ddu la phdn tir ciia B. • Phep giao Ar\B
-[x\x
& Awkx €i B].
• Phep hpp AKJ B== [x\x & A h o a c t e B\. • Hieu ciia hai tap hpp A\B=
{x I jc e A v a x ^ B}.
• Phep l^y phkn bii : Ne'u A e £ thi OEA = E \booktoan.com A ^ {X\X e E\d.x
A).
So gan dung va sai so • Cho a la gia tri dung, a la gia tri g^n dung cua a . Gia tri A^ =\a dupe gpi la sai s6 tuyet d6i ciia. s6 gdn dung a. Khi vie't a = a hieu so diing nam trong doan [a-d
-a\,
±d, ta
;a + d]. Ngucfi ta gpi d la d6 chi'nh
xac Ciia s6' g^n diing a. • Ti s6' S^ -
~
. ki hieu la S^, dupe gpi la sai sO' tucmg dO'i ciia s6
gan dung a ( t h u ^ g dupe nhan vdi 100% dd vie't du6i dang ph^n tram). • Khi thay s6' dung bcri s6' quy tron thi sai s6' tuyet d6i kh6ng vupt qua niia don vi cua hang quy tron. • Xet s6' g^n dung a ciia sG' diing a . + Ne'u a la s6' thap phan khOng nguyen, dupe vie't dudi dang chudn ma c6 k chu s6 of ph^n thap phan thi sai sO' tuyet d6\ cua a kh6ng vupt qua ™10"*,nghTala
a-i-lO"*^
2
+ Neu a la s6' nguyen dupe vie't dudi dang ehudn a = A.IO v6i A e Z va /: e N thi sai s6' tuyet ddi ciia a kh6ng vupt qua —10 , nghia la
2
2
B. DE BAI
§1. M £ N H
D^
VA M 6 N H Dfi C H O A BifiN
1.1. Trong cac cau sau day cau nao la menh dd ? Vdi eau la menh dd hay xac di»h xem menh de do diing hay sai. aj Khong dupe di qua loi nay ! b) Bay gicr la may gicf ? thiihai ke't thiic nam 1946. c) Chien tranh the giai Ianbooktoan.com
d) 4 + A: = 5. e) 1 6 c h i a 3 du 1. f) V5 la s6 v6 ti. g) Phuong trinh x^ + 3x + 5 = 0 c6 nghiem. 1.2. Neu menh dd phu dinh eiia m6i menh dd sau va xac dinh xem menh dd phii dinh d6 diing hay sai : a) P : "Phuong trinh x^ + x + l = 0 c6 nghiem". b) Q : "Nam 2000 la nam nhuan" c)R:
"327chiahetcho3".
1.3. Neu menh dd phu dinh eiia cac menh dd sau : P : "Tii giac ABCD da cho n6i tiep dupe trong du6ng tron" Q : "Tam giac ABC da cho la tarn giac can" / ? : "13 CO thd bieu didn thanh tdng ciia hai so chinh phucmg" / / : " 2^^ - 1 la mot s6' nguyen to" 1.4. Cho tam giac ABC vdi dudng trung tuye'n AM. Xet hai menh de P : "Tam giac ABC vu6ng tai A" ; Q : "Trung tuye'n AM bang niia canh BC" a) Phat bieu menh dd /* => ^ va cho bie't menh dd nay diing hay sai. b) Phat bie'u menh de P <:> Q va cho bi^t menh dd rtay dung hay sai. 1.5. Xet menh dd R : "Vi 120 chia he't cho 6 nen chia he't cho 9" Ne'u vie't menh dd R du6i dang "P => Q'\ hay neu noi dung cua cac menh dd P\aQ. Hoi menh dd R diing hay sai, tai sao ? 1.6. Cho hai menh dd P: "42 chia he't cho 5" ; Q: "42 chia he't cho 10", Phat bidu menh d6P =:> Q. Hoi menh dd nay diing hay sai, tai sao ? 1.7. Cho hai menh dd p.,-22003 - 1 la s6'nguyen t6'"; ^ : "16 la s6' chinh phuong" menh dd nay dung hay sai, tai sao ? Phat bieu menh diP ^ Q,Hdi booktoan.com
1.8. Cho hai tam giac ABC va DEF Xet cac menh dd sau P: "A = D,i
= E" ;
Q : "Tam giac ABC d6ng dang v6i tam giac DEF" Phat bidu menh diP => Q. Hoi menh dd nay diing hay sai, tai sao ? 1.9. Xet hai menh dd P : "7 la s6' nguyen l6" ; ( 2 : " 6 ! + 1 chia h^t cho 7". Phat bidu menh dd P <=> Q bang hai each. Cho bie't menh dd d6 diing hay sai. 1.10. Xet hai menh dd P : "6 la s6' nguyen t6'" ; Q:" 5\ + \ chia he't cho 6", Phat bidu menh di P <:> Q bang hai each. Cho bie't menh dd do diing hay sai. 1.11. Gpi X la tap hpp tat ca cac hoc sinh Idfp 10 of trucfng em. Xet menh dd chiia bie'n P{x) : ''x tu hoc d nha it nha't 4 giof trong mpt ngay" {x s X) Hay phat bieu cac menh dd sau bang cac cau thong thudng : a) 3x e X, P{x);
h) ^x G X, Pix);
c) 3x G X,P(x) ;
d) V x e
X,P{x).
1.12. Xet cac cau sau day : a) Ta't ca cac hoc sinh of trucfng em ddu phai hpe luat giao thong. b) Co m6t hpc sinh Idfp 12 o trucfng em c6 dien thoai di d6ng. Hay vie't eac cau d6 du6i dang " V x G X, P{xy hoac "3x s X, P(x)" va neu ro noi dung menh de chiia bie'n P(x) va tap hpp X. 1.13. Cho menh dd chiia hi€ti P{x) : "x = x'^" vdi x la s6' nguyen. Xac dinh tinh diing - sai ciia cac menh dd sau day : a)P(O); '
b)P(l);
c)P{2)\
d)/>(-l);
e)
3 A-
G Z, P{x) ;
g) \/x e Z, P{x).
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1.14. Lap menh dd phii dinh eiia cac menh dd sau : a) Vx G
R,x>x^
b) Vrt G N, «^ + 1 kh6ng chia he't cho 3. e) Vrt G N, /7^ + 1 chia het cho 4. d) 3r eQ,
r^ = 3.
1.15. Xet tinh diing menh dd do :
sai ciia cac menh dd sau va lap menh dd phii dinh eiia cac
a) 3r G Q, 4r^ - 1 = 0. b) 3n G N, n^ + 1 chia het cho 8. c)Vx
eR,x^
+
x+\>0.
d) V« G N*, 1 + 2 + ... + n khong ehia he't cho 11. 1.16. Cho menh dd ehiia bie'n P(x) : "x thich m6n Ngft van", trong do x \iy gia tri tren tap hpp Xcac hpc sinh ciia trudng em. a) Diing ki hieu I6gic de didn ta menh dd : "Mpi hpc sinh cua trucmg em ddu thieh m6n Ngu van." b) Neu menh dd phu dinh ciia menh dd tren bang ki hieu logic r6i didn dat menh dd phii dinh do bang cau th6ng thucmg. 1.17. Cho menh dd chiia bie'n P{x) : "x da di may bay", trong do x \&y gia tri tren tap hpp X eac eu dan eiia khu phd (hay xa) em. a) Dung ki hieu logic dd didn ta menh dd : "Co m6t ngu6i ciia khu ph6' (hay xa) em da di may bay'' b) Neu menh dd phu dinh eua menh de tren bang ki hieu I6gic r6i didn dat menh dd phii dinh bang cau th6ng thudng.
§2. A P D U N G MfiNH Bt VAO SUY LUAN TOAN HOC 1.18. Phat bieu va chiing minh cac dinh If sau : a) Vn G N, n" ehia he't cho 3 => n chia he't cho 3 (gen y : Chiing minh bang phan ehiing). b) V« G N, n^ chia he't cho 6=> n chia het cho 6. booktoan.com
1.19. Cho eac menh dd ehiia bien P{n) : "n la s6' chan" va Q{n) : "In + 4 la s6' chan" a) Phat bidu va chimg minh dinh Ii Vn G N , P{n) => Q{n). b) Phat bieu va chiing minh dinh If dao cua dinh If tren. c) Phat bidu gpp dinh li thuan va dao bang hai each. 1.20. Cho cac menh de chiia bie'n P{n) : "n chia he't cho 5" ; Q{n) : "n ehia he't 2
2
•
cho 5" va R{n): "n + 1 va n - 1 deu khOng ehia het cho 5" Sii dung thuat ngfi "didu kien e^n va dii", phat bidu va chiing minh cae dinh li dudi day : a) V/7 e N, P{n) <=> Q(n). b) V/7 G N, P{n) ^
R{n).
1.21. Cho eac s6' thuc ay,a2,—,a^^. Gpi a la trung binh e6ng ciia ehung ai + ... + a„ a =— -• n Chung minh (bang phan chiing) rang : ft nhS^t m6t trong cac s6' a^,a2,...,a„ se Idn hon hay bang a. 1.22. Sir dung thuat ngu "didu kien du" dd phat bidu cac dinh li sau : a) Ne'u hai tam giac bang nhau thi ehiing d6ng dang v6i nhau. b) Ne'u m6t hinh thang eo hai dudng cheo bang nhau thi no la hinh thang can. c) Ne'u tam giac ABC can tai A thi ducfng trung tuyen xuat phat tir dinh A cung la ducfng cao. 1.23. Sir dung thuat ngiJ "dieu kien e^n' de phat bieu eac dinh If sau : a) Ne'u mpt sd nguyen duong le dupe bieu didn thanh tong ciia hai sd ehfnh phuofng thi s5' do phai c6 dang Ak + 1 (^ e N). b) Ne'u m, n la hai s6' nguyen ducrng sao cho nr + n^ la m6t so chinh phuong thi m.n ehia het cho 12.
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1.24. Hay phat bidu va ehiing minh dinh If dao ciia dinh If sau (ne'u eo) r6i sir dung thuat ngfl didu kien "c^n va dii" dd phat bidu g6p ca hai dinh If thuan va dao : Ne'u m, n \a hai s6 nguyen duong va m6i s6' ddu ehia he't cho 3 thi t6ng m^ + r? cung chia h^t cho 3.
§3. TAP HOP VA CAC PHEP TOAN TRfiN TAP HOP 1.25. Cho A la tap hpp cac hinh binh hanh c6 bO'n goe bang nhau, B la tap hpp eac hinh chii nhat, C la tap hpp cac hinh thoi va D la tap hpp cac hinh vu6ng. Hay neu m6i quan he giiia cac tap noi tren. 1.26. Cho^ = { 0 ; 2 ; 4 ; 6 ; 8 | , f i = { 0 ; 1 ; 2 ; 3 ; 4} vaC = {0 ; 3 ; 6 ; 9|. a) Xac dinh (A u fi) u C va ^ u (B u C). Co nhan xet gi vd ke't qua ? b) Xac dinh (A n B) n C va A n (B n C). Co nhan xet gi vd ke't qua ? 1.27. Cho A - {0 ; 2 ; 4 ; 6 ; 8 ; 10}, S = {0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6| va C = | 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10}. Hay tim a) A n (B n C) ;
b) A u (B u C) ;
c) A n (B w C) ;
d) (A o B) n C ;
€){Ar\B')vjC. 1.28. Ve bidu d6 Ven thd hien cac phep toan sau cua eac tap A, B va C : a) A n (B u C);
b)(A \B)KJ{A\C)VJ{B\
C).
1.29. Co thd noi gi vd eac tap A va B neu eac ding thd'e tap hpp sau la diing : a ) A w B = A;
\y) Ar^B = A\
C)A\B = A;
d ) A \ B = B\A.
1.30. Lieu CO thd ke't luan A-B a)A^C = BwC;
dupe kh6ng ne'u A, B va C la cac tap thoa man \>) Ar\C = Br\C
1.31. Vdi m6i tap A c6 m6t s6' hihi han p h ^ tir, kf hieu lAt la sd p h ^ tii ciia tap A. sap xe'p cac s6' sau day theo thu: tu tang d^n : a) lAl, lAw BI, lAnBl ;
b) 1A\BI, \A\ + IBI, lA^Sl. booktoan.com 11
132. Cho t a p A = { x G R | 2 2} thanh hpp cac nvra khoang. 1.34. Chimg minh rang V6 la sd v6 ti. 1.35. Cho A = {x e R |
^ > 2 } vaB = U G ]R| Lc - II < U-Hay tim I jv - 2 I
A^ B va An B. 1.36. ChoA=^ {;c G R | U - II < 3} vaS = |X e R | lx +21 > 5). Hay timA n B .
§4. s6 GAN DUNG VA SAI S6 99 Qung ae. xap xi, vz. /17 — —, 1.37. Trong hai so —-, —- diing de xap xi V2.
,.^ ^.,,,.. ^^.p
a) Chijmg to rang — xa'p xi V2 t6t hon. 99
r
•>
-5
b) Chimg minh rang sai sd tuyet ddi cua — so vdfi V2 nho hon 7,3.10 355 1.38. Cae nha toan hpc da xap xi sd n boi sd —— Hay danh gia sai sd tuyet ddi bie't 3,14159265
(0 < X < 1). Gia sir ta ISiy s6 a = \ - Jt lam gia tri gdn
diing cua a. Hay tfnh sai s6' tuong ddi cua a theo x.
BAI TAP 6 N TAP CHl/ONG I 1.42. Xet cae menh de ehiia bien sau : P(x) : "x la mot ki su", Q{x) : "x la mot ngudi ed tay nghd" va Rix) : 'x la mot ngudi ed thu nhap cao" Goi X la tap hpp toan the loai ngudi. Hay didn dat bang ldi eac menh dd sau : booktoan.com 12
a) \/XGX,P(X)^Q{X)
;
b)
VxGX,Qix)=^R(x)-
e) \/xe
X,P(x)^
R(x).
1.43. Lap menh dd phii dinh ciia mfnh dd "V« e N, rt^ + rt + 1 la sd nguyen td". Menh dd phii dinh dd diing hay sai ? 1.44. Hay phat bidu va ehung minh dinh If dao eiia dinh If sau (n^u cd) rdi sir dung thuat ngu didu kien c^n va du de phat bidu gpp ca hai dinh If thuan va dao : Ne'u hai sd duang bang nhau thi trung binh edng va trung binh nhan ciia ehiing bang nhau. 1.45. Chung minh cac dinh If sau bang phuong phap phan ehung : a) Trong mot tii giac I6i phai cd ft nha't mot goc khdng nhpn (Idn hon hay bang gde vu6ng) va ed ft nha't mdt gde khong tii (nho hon hay bang gde vudng). b) Ne'u ;t va J la hai sd thue vdix ^ - 1 va _y ^^ -I thi x + y + xy ^ - 1 . 1.46. Cho menh dd chiia bie'n P(m ; n) : "n ehia he't cho m" vdi m la sd nguyen duong, n la cae sd tu nhien. Xac dinh tfnh dung - sai ciia cac menh dd sau : a)/'(4;5); b) ^(2 ; 4 ) ; c) V« G N, Vm G N*, P(m \n)\ e) 3n &n,\fm^
d) 3m
G N*,
\fn
G N,
P{m ; n);
N*, P{m ; n).
1.47. Cho A va B la hai tap hpp h&u han. Kf hieu lAI la sd phdn tii cua tap hpp A. a) Chung minh rang neu A n B = thi lA w BI = lAl + IBI. b) Chiing minh rang B^u (A \B) = A u B v a B n {A\B) = 0. c) Chung minh ring A-{Ar\B)
u (A \ B).
d) Tijr dd suy ra edng thiic sau \A u BI = lAl ^ IBI - lA n B\. 1.48. Cho A = {;t e R I U - l l > 3 } v a 6 = U G R l lx + 2 l < 5 } . T i m A n B . 1.49. Ngudi la goi m6t sd hOu ti r cd dang r = — la sd hiJii ti nhi phan. 2" Bie't rang trong mdi khoang tuy y ddu cd ft nha't mdt sd huu ti nhi phan. Qidng minh rang trong mdi khoang ba't ki ddu cd ft nheit 100 sd huu ti nhi phan.
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13
M6t each t6ng quat ehung minh rang : Cho m6t sd nguyen ducfng M Idn tuy y. Khi do, trong mdi khoang tuy y ddu ed ft nh^t M s6 hiiu ti nhi phan. 1.50. Gia sir;c la mdt gia tri gdadung ciia v5 . Xet sd a =
x+2
. Chiing minh rang
\a'j5\<\x-yf5\. tire la ne'u la'y a la gia tri g^n diing ciia v5 thi ta dupe dd ehfnh xac cao hon la la'y x.
Gldl THifiU MOT S 6 CAU H O I TRAC NGHlfiM K H A C H QUAN 1.51. Trong cdc menh dd dudi day menh dd nao ddng, menh dd nao sai ?
b) Vrt e N, n^ + 1 khdng chia h^t cho 3.
Qtiung
Dsai Dsai
c) Vn e N, «^ + 1 chia h^t cho 4.
n^^ung
Dsai
d)3rG
n*Jung
Dsai
a) V;c G
Ddung
R,x>x^.
= 3.
Trong cdc bdi tit 1.52 din bdi 1.54 hay chon phuang an tra ldi diing trong cac phuang an da cho. 1.52. Cho cac cau sau : a) Hai Phdng la mdt thanh phd d Midn Nam. b) Sdng Hdng chay qua thii dd Ha N6i. e) Hay tra ldi cau hoi nay ! d) 2 + 37 = 39 ; e) 5 + 40 = 70 ; g) Ban cd rdi tdi nay khdng ? h) A: + 2 = 11 ; Sd cau la menh di trong cae eau tren la (A)l: (B)2; (C)3; (D) 4 ;
(E) 5.
1.53. Cho menh dd chiia bie'n P{x) : "jc + 15 < x^" \dix la sd thue. Menh dd diing la menh dd : (A) P{0); (B) B(3); (C) B(4); (D) Pi5).
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14
1.54. Cho menh dd " Vx G R, x^ + x + 1 > 0". Menh dd phii dinh eiia menh dd tren la : (A) Vx G R, ;c^ + X + 1 < 0 ;
>
(B) Vx G R, x^ + ; c + 1< 0 ; (C) Khong ton tai X G R ma x^ + x + 1 > 0 ; (D) 3 x G R, x^ + x + 1< 0. 1.55. Trong cae menh de sau day menh dd nao khdng la dinh If: (A) V/i G N, n^\2 =^ n':2 ;
(B) VM e N, n^: 3 => « : 3 ;
(C) Vrt e M, «^ ; 6=^ rt ; 6 ;
(D) \/n e N, n^': 9^
n : 9.
1.56. Trong eac menh dd sau day menh dd nao la mfnh dd diing. (A) Vx G R, X > - 2 => x^ > 4 ;
(B) Vx G R, x > 2 => x^ > 4 ;
(C) Vx G R, x^ > 4 => X > 2 ;
(D) Vx G R, x^ > 4 => x > - 2 .
Trong cdc bdi tiJC 1.57 den 1.63, hay chon phuang an tra ldi diing trong cdc phuang an dd cho. 1.57. Trong cac sd dudi day, gia tri g^n diing ciia V65 - v63 vdi sai sd tuyet ddi be nha't la : (A) 0,12 ;
(B) 0,13 ;
(C) 0,14 ;
(D) 0,15.
1.58. Cho tap A = {-1; 0 ; 1 ; 2}. Khi dd ta cung eo : (A) A = [-1 ; 3) n N ;
(B) A = [-1 ; 3) n Z ;
(C) A = [-1 ; 3) n N* ;
(D) A = [-1 ; 3) n Q.
1.59. Cho doan M = [-4 ; 7] va tap A' = (-oo ; - 2 ) ^ (3 ; +oo). Khi dd M n A^ la
1.60.
(A) [-4 ; - 2 ) w (3 ; 7] ;
(B) [-4 ; 2) L; (3 ; 7 ) ;
(C) (-00 ; 2] u (3 ; +^);
(D) (-oo ; - 2 ) w [3 ; +oo).
Cho hai tap hpp
A = {XGR
| X + 3 < 4 + 2X};
S = {xG R 1 5 x - 3 < 4 x - 1}. Ta't ca cdc sd tu nhien thudc ca hai tap A va B la (A) 0 va 1 ; (C) 0 ;
'
(B) 1 ; (D) Khdng cd sd nko.
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1.61. Cho cac nira khoang A = (^co ; -2] ; B = [3 ; +oo) va khoang C = (0 ; 4) Khi do tap (A u B) n Cla (A)
{XGRI3
;
(B) {x G R I x < - 2 hoaex > 3} ;
(C) | x e E I 3 < x < 4 } ;
(D) {x G R l x < - 2 hoac x> 3}.
1.62. Cho cac khoang A (-2 ; 2) ; B = (-1 ; +co) va C = -oo ; - . Khi dd giao V
LJ
Ac\Br\C\a (A) X G R I -1 < X < i
;
(B) Ix G R 1 -2 < X < 1} ;
(C) X G E I -1 < X < 11 ;
(D) Ix G R I -1 < X < i | .
1.63. Cho sd thuc a < 0. Didu kien eSn va dii de hai khoang (-co ; 9a) va
4
V
— ; + 00 CO giao khae tap rong la a
I
(A)-|<«<0;
(B) - | < a < 0 ;
(C) - 4 < a < 0 ;
(S>)-\
4
4
C: DAP SO - HUONG DAN - LOI GiAl 1.1. Cac cau e) va f) la menh dd diing. Cac cau e) va g) la menh dd sai. cac eau edn lai khong phai la menh dd. 1.2. a) P : "Phuong trinh x^ + x + 1 = 0 vd nghiem" P la menh dd diing. b) Q : "Nam 2000 khdng phai la nam nhuan" Q la menh dd sai. c) R : "Sd 327 khdng ehia he't cho 3" R la menh dd sai. 1.3. a) P "Tii giac ABCD da cho khdng ndi tie'p dupe trong dudng trdn'\ b) Q "Tam giac ABC da cho khdng phai la tam giac can'' c) R : "Sd 13 khdng thd bidu didn thanh tdng ciia hai sd chinh phuong" d) a : "Sd 2^^ - 1 khdng la sd nguyen td"
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1.4. a) "Ne'u tam giac ABC da cho vudng tai A thi trung tuydn AM bang niia canh BC. Menh dd nay diing. b) "Tam giac ABC da cho vudng tai A ne'u va chi ndu trung tuy^n AM bang niia canh BC" Menh dd nay diing. 1.5. P: "120 chia het cho 6" Q : "120 ehia he't cho 9" Menh dd R sai vi P diing Q sai. 1.6. "Do 42 chia he't cho 5 nen no ehia he't cho 10" Menh dd nay diing vi P la menh dd sai (cho dii Q diing hay sai). 1.7. "Ne'u 2^°^^ - 1 la sd nguyen td thi 16 la sd chinh phuong" Menh de nay diing vi Q la menh dd diing (cho dii P dung hay sai). 1.8. "Ne'u A = S, B = £ thi tam giac ABC ddng dang vdi tam giac DEF" Menh dd nay diing. 1.9. "7 la sd nguyen td neu va chi ne'u 6! + 1 ehia he't cho 7". "Didu kien edn va du dd 7 la sd nguyen td la 6! + 1 ehia h^t cho 7" M6nh dd diing vi ca hai menh di P \aQ ciing diing. 1.10. "6 la sd nguyen td ne'u va chi ne'u 5! + 1 chia he't cho 6" "6 la sd nguyen td khi va ehi khi 5! + 1 chia he't cho 6". Menh dd dung vi ca hai menh di P vaQ ddu sai. 1.11. a) "Cd mdt ban hpc d Idp 10 d trudng em tu hpe ft nha't 4 gid trong mdt ngay'' b) "Mpi hpc sinh Idp 10 d trudng em tu hpc ft nha't 4 gid trong mdt ngay". c) "Cd mdt ban Idp 10 d trudfng em tu hpe ft hofn 4 gid trong mdt ngay" d) "Mpi hpe sinh Idp 10 d trudng em tu hpe ft hon 4 gid trong mdt ngay". 1.12. a) " Vx G X,P{x)" trong dd X la tap hpp ta't ca cac hpc sinh d trudng em, P{x) la menh dd chiia bie'n : "x hpc luat giao thdng" b) 3x e X,P{x)" trong dd X la tap hpp ta't ca eac hpc sinh Idp 12 d trudng em, P{x) Ik menh dd chu:a bie'n : "x ed dien thoai di ddng" 1.13. a) Menh dd dung ; e) Menh dd sai; e) Menh dd diing ;
b) Menh dd ddng ; d) Menh dd sai; g) Menh dd sai.
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17
1.14. a) 3x G R, X < x^ b) 3/7 G N, rt^ + 1 ehia he't cho 3. c) 3tt e N, «^ + 1 khdng chia he't cho 4" d) Vr G Q, r^ ?t 3. 1
•>
1.15. a) Menh dd dung vi vdi / = -- thi 4r'^ - 1 = 0. Menh dd phii dinh la " V r G Q, 4r^ - 1 ? ^ 0 "
b) Menh dd sai. Ta ehung td menh dd phii dinh "\/n e N, «^ + 1 khdng ehia he't cho 8" la diing. That vay, ne'u n la sd chan thi n^ + 1 la sd le nen khdng ehia het cho 8. Neu n la sd \e,n = 2k+\{ke N) thi n^+\=
4k{k + 1) + 2 ehia 8 du 2 ( vi k{k + 1) la sd chan).
e) Menh dd diing. Menh dd phu dinh "3x G R, x^ + x + 1 < 0" d) Menh dd sai. Ta ehiing to menh dd phii dinh "3n G N , l + 2 + - - - + n chia he't cho 11" la diing. That vay vdi n = 11 thi 1 + 2 + ••• + 11 = 66 chia he't cho 11. 1.16.
a)
VXGX,B(X).
b) 3x G X,P{x),
nghia la "Cd mdt ban hpc sinh ciia trudng em khdng
Ihi'eh mdn Ngii van". 1.17. a ) " 3 x G A',B(x)" b) Menh dd phu dinh : "Vx e X,P{x)" nghia la : "Mpi ngudi trong khu phd (hay xa) em ddu chua di may bay" 1.18. a) "Ne'u n \a sd tu nhien sao cho n ehia he't cho 3 thi n cung ehia hdt cho 3", Ta chiing minh bang phan chiitig. Gia su tdn tai « G N de n ehia het cho 3 nhung n khdng chia hdt cho 3. Ne'u « = 3A: + 1 (/: G N) thi n^ = 3k{3k + 2) + 1 khdng chia het cho 3. Neu n = 3k-i {k e N)tlu n^ = 3k{3k - 2) + 1 khdng chia he't cho 3. b) "Ne'u n la sd tu nhien sao cho n^ chia hd^t cho 6 thi n cung chia he't cho 6". That vay ndu n^ ehia he't cho 6 thi n^ la sd chan, do dd n la sd chan, tiic la n ehia he't cho 2. Vi n^ chia he't cho 6 nen nd chia hdt cho 3. Theo cau a) didu nay keo theo n chia he't cho 3. Vi n chia he't cho 2 va 3 nen n chia he't cho 6. booktoan.com 18
2-BTDS10,NC - B
1.19. a) Phat bidu : " Vdi mpi sd tu nhien «, ne'u n chan thi 7n + 4 la sd chan." Chiing minh. Ne'u n chan thi In chSn. Suy ra 7n + 4 chan vi tong hai sd chan la sd chan. b) Dinh If dao : "\fn G N , Qin) => P{n)" tiic la "Vdi mpi sd tu nhien n, ne'u 7« + 4 la sd chan thi n eh^n" ChUng minh. N^u In + 4 = m chan thi In = m - 4 chin. Vay In chan nen n chan. c) Phat bidu gdp hai dinh If thuan va dao nhu sau : "Vdi mpi sd tu nhien n, n chan khi va ehi khi 7n + 4 chan" hoac "Vdi mpi sd tu nhien n, n chan ne'u va chi ne'u 7/7 + 4 chan". 1.20. a) Phat bidu nhu sau : "Didu kien edn va dii dd sd tu nhien n chia he't cho 5 la rt chia het cho 5" Chi/tng minh. Ne'u n = 5k [k e N) thi n^ = 25k^ chia he't cho 5. Ngupe lai, gia sir /z = 5jt + r vdi r = 0, 1, 2, 3, 4. Khi dd n^ = 25^^ + lOkr + r^ chia he't cho 5 nen /• phai ehia he't cho 5. Thii vao vdi r = 0, 1, 2, 3, 4, ta tha'y chi cd vdi r = 0 thi r cho 5.
mdi ehia he't cho 5. Do do n = 5k t\tc la n ehia het
b) Phat bidu nhu sau : "Didu kien cdn va dii dd sd tu nhien n ehia he't cho 5 la ca /7^ - 1 va /7^ + 1 ddu khdng chia het cho 5" Chimg minh. Ne'u n ehia he't cho 5 thi n^ - 1 chia 5 du 4 va /7^ + ! chia 5 du I. Dao lai, gia sir /? - \ va n + 1 ddu khdng ehia he't cho 5. Gpi /• la sd du khi chia n cho 5 (r == 0, 1, 2, 3, 4). Ta c6 n = 5k ^ r {k ^ N). Vi n^ = 25/t^ + lOkr + r^ nen suy ra ca r^ - 1 va r^ + 1 ddu khdng chia he't cho 5. Vdi r = 1 thi r^ - 1 = 0 chia hdt cho 5. Vdi r = 2 thi r^ + 1 = 5 chia he't cho 5. Vdi /• = 3 thi r^ + 1 = 10 ehia het cho 5. Vdi /• = 4 thi /-^ - 1 = 15 chia het cho 5. Vay ehi cd the r = 0 tiic \an-5k
hay n chia he't cho 5.
1.21. Chiimg minh bang phan ehung nhu sau : Gia su trai lai ta't ca eac sd a^,a2....,a,^ ddu nho hon a. Khi dd ai + ^2 + • • • + a„ < na suy ra a = -^
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^ < a. Mau thuln. 19
1.22. a) Didu kien dii dd hai tam giac ddng dang la ehiing bang nhau. b) Bi mdt hinh thang la hinh thang can, didu kien dii la hai dudng cheo ciia nd bang nhau. c) Didu kien dii dd dudng trung tuye'n xua't phat tijf A eua tam giac ABC vudng gde vdi BC la tam giac do can tai A. 1.23. a) De mdt sd nguyen duong le bidu didn thanh tdng cua hai sd ehfnh phuong didu kien edn la sd dd ed dang 4 ^ + 1 . b) Cho m, n la hai sd nguyen ducfng. Didu kien e^n di m + n la sd ehfnh phuong la tfeh mn chia he't cho 12. 1.24. Dinh If dao : "Ne'u m, n \a hai sd nguyen duong \k m + n ehia he't cho 3 thi cam van ddu chia he't cho 3" Chifng minh. Ne'u mdt sd khdng chia he't cho 3 va sd kia ehia he't cho 3 thi rd rang t6ng binh phuong hai sd do khdng chia h^t cho 3. Gia sis m va n ddu khdng chia he't cho 3. Ne'u m = 3k + 1 hoac m = 3k + 2 ta ddu cd /M^ehia 3 du 1. Thanh thir m^ + n^ chia 3 du 2. Vay n^u m^ + n^ chia he't cho 3 thi ehi ed thd xay ra kha nang cam van ddu chia he't cho 3. vay : Didu kien c^n va dii dd /n^ + n^ ehia he't cho 3 (/n, /z G N*) la ca m va n ddu ehia hdt cho 3. 1.25. TaedA = B ; D (z B ^ A ; D czC ; D = BnC. 1.26. a) A u B = { 0 ; l ; 2 ; 3 ; 4 ; 6 ; 8 } , ( A w B ) u C = {0 ; 1 ; 2 ; 3 ; 4 ; 6 ; 8 ; 9}. BL;C={0;1;2;3;4;6;9},AW
(BuC)={0;l;2;3;4;6;8;9}.
Tacd (AwB)^ C = Au (BuC). b ) A n 6 = { 0 ; 2 ; 4 } , ( A n B ) n C = {0}. B n C = {0;3},A n (B n C ) = {0}. Ta cd (A nB) n C = A n (B r\C). Chu y : Cd thd chiing minh dupe rang cac ding thiic tren ludn diing y6i, A, B, C la ba tap hpp ba't ki. 1.27. a) A n (B nC)= {4;6} ; b)A ^ (B uC)= { 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10}.
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c)A n (B yuC) = A. d)A uB=
( 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 8 ; 10}.
vay (A wB) n C = |4 ; 5 ; 6 ; 8 ; 10}. e)A n B = { 0 ; 2 ; 4 ; 6 } . Vay (A nB) L ^ C = { 0 ; 2 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10}, 1.28. a)
b)
Phctn gach cheo la hinh bi^u diin A n (B u C)
Ph^n gach cheo la hinh bidu difin (AsB) u (A\C) u (S\C)
Hinh l.l
Hinh 1.2
1.29. a) Ne'u A ^ B = A thi 6 la tap con ciia A vi theo dinh nghia ta ludn cd B(^A^B. Di kiem tra rang didu ngupe lai ciing dung. Vay A w B = A neu va chi ne'u B la tap con eiia A. b) Ne'u A n B = A thi A la lap con cua B vi theo dinh nghia ta ludn ed AnB c B . c) Neu A \ B = A thi hai tap A va B phai khong giao nhau. That vay, neu ton tai X e A va X e B thi do A =^ A \ B nen x G A\B. Suy ra x khdng thupc B (mau thutn). Ngupe lai bang each ve bidu dd Ven dd tha'y ne'u AnB = 0 thi A \ B = A cung diing. Vay A \ 6 ^ A nduva chi ndu A n B = 0 . d) Neu A \ B = B \ A thi A = B. That vay ne'u A ^ B thi phai cd mot phdn tir ciia tap nay nhung khong thupc tap kia, chang han x e A va x ^ B suy r a x G A\B nen x G B \ A do d d x e B v a x ^ A (mau thuan). Dd kiem tra rang didu ngUpe lai ciing diing. Vay A \ B = B \ A n^u va ehi neu A = B . booktoan.com .21
1.30. a) Khong. Chang han A - {1 ; 2 ; 3 ; 4}, B = {1 ; 2}, C = {3 ; 4 ; 5}. A =^ B nhung
A u C^B u C= {1 ;2;3;4;5}. b) Khdng. Chang han A - {1 : 2 ; 3 ; 4}, B = {3 ; 4}, C = {3 ; 4 ; 5}. Ta c6 A ^ B nhung A n C = B n C= { 3 ; 4 } . 1.31. a) \A n B\, \A\, lA u BI ; b ) l A \ B U A wBl, lAl + lBl. 1.32.A = ( 2 ; 3) u ( - 3 ; - 2 ) . 1.33.A-[2,+oo)LJ ( - 00 ; - 2 ] . 1.34. Chutig minh bang phan chiing. Gia su v6 = — la mot sd hiiu ti trong dd a, h la hai sd nguyen duong va (a, b) = \. Suy ra 6/? = a Vay a chia he't cho 2 va chia het cho 3. Suy ra a chia he't cho 2 va chia he't cho 3 tiic la a chia het cho 6. Dat a = dk {k e N*). Thay vao ta dupe 6b^ = 36k^ hay b ~ 6k Lf luan tuong tu nhu tren ta suy ra b chia het cho 6. vay a va b CO ude ehung la 6. Didu nay mau thuln vdi gia thie't a, b khdng ed udc ehung Idn hon 1. 1.35.TacdA = {x G M l O < l x - 2i < 0 , 5 } , s u y r a A = (1,5 ; 2,5)\{2}. Dd tha'y B-(0;2). Tii do A u B = (0;2,5)\{2} vaA n B = (l,5 ; 2). 1.36. T a e d A = ( - 2 ; 4 ) v a B = (3,+oo) u ( - o o ; - 7 ) . Vay A n B = (3 ; 4). r 99 17 99 r 1.37. a) Ta ed V2 < — < — do dd — xap xi V2 tdt hon 99 b ) T a c d — = 1,414285714... < 1,414286, V2 ^ 1,4142135... > 1,414213. 99 u oo ddod O u << - ^ - V2 < 1,414286- 1,41421*3 « 0,000073. D
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1.38. Ta cd (su dung may tfnh bo tiii) : 355 113
3,14159292... < 3,14159293.
355 Do vay 0 < — - n < 3,14159293 - 3,14159265 ^ 0,00000028. Vay sai sd tuyet ddi nho hon 2,8.10"^ 1.39, Ta ed AL^ = BLLD = 2, do dd AL =^2 . Lai ed BD = 3, suy ra dien tfeh ciia hinh e h u n h a t l a 3 V 2 =3.1,41421356... «4,24264... «4,24. 1.40. Chii sd 3 (hang phan tram) la chu" sd chae do 0,00312 < 0,005. Do dd C cd 3 chu" sd chae (d hang don vj, hang phdn chuc va hang phdn tram). .
1
Hinh 1.3
..2
. Sai sd tuong ddi la S^ - —a~ I+ v " 1 - *;^ 1.42. a) Ne'u mdt ngudi la ki su thi ngudi do ed lay nghd. 1.41.
^' = rrx
(1 - x)
_
1 - x^
b) Ne'u mdt ngudi khdng cd lay nghd thi ngudi dd khong cd thu nhap cao. c) Ne'u mot ngudi la fci su thi ngudi a'y cd thu nhap cao. 1.43. Menh de phii diiih la "3n e N, n^ + /7 + 1 khdng la sd nguyen td" Menh dd phii dinh dung. Vf du vdi /? = 4 thi /i~ + /7 + 1 = 2 1 chia he't cho 3 nen la,hpp sd. 1.44. Dinh li dao : "Ne'u hai sd duong a, b cd trung binh edng va trung binh nhan bang nhau thi chiing bang nhau.' a +b Chitng minh. Gia sir a, b la hai sd duong sao cho = ^ab Khi dd
a + b-2yfab
=^0 c> (Ja - Sf=0
::^ a = h.
vay didu kien c^n va dii de hai sd duong bang nhau la trung binh epng va trung binh nhan cua ehung bang nhau. 1.45 a) Gia sii ea bdn gde ddu nhpn. Khi dd tdng cua bdn gde cua tu" giac se nho hon360" (mau thuan). Tuong tu gia sii ca bdn gde ddu tii. Khi dd t6ng ciia bon gde ciia lii giiic se Idn hon 360° (mau thuan).
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b) Gia siix + y + xy = - 1 . Suy rax + y + xy + 1 = (x + l)(y + 1) = 0. vay phai ed hoac x = - 1 hoac y = - 1 (mau thu^n). 1.46. a) Menh dd sai. b) Menh dd diing. e) Menh dd sai. d) Menh dd diing (vi v6im= I thi n chia he't cho m vdi mpi n). e) Menh di diing (vi v6i n = 0 thi n ehia h^t cho m vdi mpi m). 1.47. a) Hien nhien. b) De tha'y bang each ve so d6 Ven. c) Dd tha'y bang each ve so dd Ven. d) Ta cd lA uBl = IB! + lA \Bl, (do cau a) va b)).
(1)
Lai cd A = (A \B) u(A n B) ( do c)) thanh thir UI = IA\BI + IA n BI.
vay IA\BI = IAI-IA n BI.
(2)
Thay (2) vao (l)ta dupe IAWBI = IAI + I6l-IA n BI. 1.48. Tae6A = (4;+oo)w (-oo ; - 2 ) ; B = ( - 7 ; 3). Vay A n B = ( - 7 ; - 2 ) . 1.49. Gia sir (a ; b) la mdt khoang ba't ki. Ta chia (a ; b) lam 100 khoang eon rdi nhau. Theo nhan xet tren trong mdi khoang con dd ddu cd chiia mdt sd hixu ti nhi phan. Cac sd hiJu ti nhi phan nay khae nhau do cac khoang con khong giao nhau. Vay (a ; b) chiia ft nha't 100 sd hiiu ti nhi phan. Md rOng : Ta chia khoang (a ; b) lam M khoang eon rdi nhau. Theo nhan xet tren trong mdi khoang con dd ddu cd chiia mot sd hiiu ti nhi phan. Cac sd hUu ti nhi phan nay khae nhau do eac khoang con khdng giao nhau. Vay {a ; h) chiia ft nha't M sd huu ti nhi phan. 1.50. Dat M - X - V5 va V = fl - V5 Ta cd rz -=^~^524
2x + 5 - xVs - 2V5 (2 - %/5)(x - VS) (2 - V5)M J ^ 772" - ;, + 2 •
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vay a-yf5\=\v\= \u\^^~^ <^^~^ \u\<\u\ = x-yf5 • I x+2 2 1.51. Cau a) la menh dd sai. cau b) la menh dd diing. That vay neu /7 = 3)t thi /7^ + 1 = 9A:^ + 1 chia 3 du 1. Ne'u /7 = 3jt + 1 thi /i^ + 1 = 9k^ + 6Jt + 2 chia 3 du 2. Neu n =3k + 2thin^ +\ = 9k^ + 12/: + 5 chia 3 du 2. cau e) la menh dd sai. That vay ne'u n = 2k thi n^ + \ = 4k^ + \ chia 4 du 1. N^u n = 2k+lthi n^ + \ =4k^ + 4 ^ + 2 chia 4 du 2. Cau d) la menh dd sai do V3 la sd v6 ti. 1.52. Phuong an (D). (Cae cau a), b), d), e) la cac menh dd). 1.53. Phuong an (D). 1.54. Phuong an (D). 1.55. Cae menh dd (A), (B) va (C) la menh dd diing Menh de (D) la sai vi vdi n = 3 thi 3^ = 9 ehia h^t cho 9 nhung 3 khdng chia he't cho 9. Do dd menh dd (D) khdng phai la dinh If. Vay ta chpn phuong an (D). 1.56. (A) la menh dd sai. That vay vdi x = 0 thi 0 > - 2 nhung 0 < 4. (B) la menh dd diing. (C) la menh dd sai. That vay vdi x = - 3 thi (-3)^ = 9 > 4 nhung - 3 < 2. (D) la menh dd sai vi chang han, khi x = - 3 thi (-3)^ > 4 nhung -3<-2. Do dd ta chpn phuong an (B). 1.57. Sii dung may tfnh cho ta V65 - >/63 «0,I25003815... Do dd ta chpn phuong an (B). 1.58. Phuong an (B). 1.59. Phuong an (A). 1.60. Phuong an (A). 1.61. Phuong an (C). 1.62. Phuong an (D). 1.63. Phuong an (A).
I
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25
uang II *'
HAM SO
A.
N H Q N G KIEN
THLfC CAN NHO
Ham so' Trong bang sau day, y ~ f(x)
\a mot ham sd vdi tap xac dinh S), K la
mdt khoang (niia khoang hay doan) nam trong 3). Tinh chat cua ham so
The hien qua do thi
yo = fi^) (vdixothudcy^).
Diem (XQ ; yp) thude do thi cua ham sd. y 1 1 1
Ham s d / d 6 n g bie'n tren K : VXi,X2 eKlX^
o
/(Xi)
a
b
X
Tren K, dd thi ciia ham s d / d i len (theo chidu tang ciia ddi sd).
Ham sd/nghich bien tren K:
I
I
O
Vxi,X2 G /f :X| /(xi)>/i;x2).
Tren K, dd thi eua ham s d / d i xudng (theo chidu tang ciia ddi so').
booktoan.com 26
yk
Ham sd/khdng ddi tren K : y = m{m\a hang sd).
o
b
X
W thi eua ham s d / n l m tren dudng thang song song (hoac triing) vdi Ox.
y =f{x) la ham sd chan : Vx € 3): -X G ® va/(-x) =/(x).
Dd thi cua ham so fed true ddi xiing la true tung Oy.
y =f(x) la ham sd le : Vx e ^"^: -X G y^' va/(-x) = -fix).
B6 thj eua ham s d / e d tam ddi xung la gde toa dp O.
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Ham so bac nhat • Ham sd cho bdi bidu thiic y = ax + b (a^^tQ). Tap xac dinh : • Bang bien thien : X
-00
y = ax + b {a>0)
^ ^ ^ —00
+00
-00
X
+00
+00
y = ax + b ia<0)
^"^'''''^
~-*' - 0 0
• D6 thi ciia ham s6 y = ax + b {a ^ 0) la dudng thing cd he sd gde bang a, eat Ox tai ( — ; 0) va cat Oy tai (0 ; b). a • Ne'u (di) va (J2) la hai dudng thing phan biet ed he sd gde la Oj va ^2 thi; (Ji) 7/(^2)
<^
(rfj) eat ((^2) <=> i?i =?^ «2 ' d^ X dn <^ a^.a2 - - 1 .
Ham so bac hai • Ham sd cho bdi bieu thiic y^ax
+ fex + c (« ^ 0). Tap xac djnh : M
• Bang bie'n thien :
.
^
b 2a
-00
2 y = ax •>t-bx + c (fl>0)
f^'^
" \
\
A/ 4a
X
+00
2
y~ax
—00
+00
A / "4^\
L.
+ bx + c
(«<0)
h 2a
/ —00
\ -00
» D6 thi ciia ham sd y ~ ax •¥ bx + c {a ^ H) \a. parabol cd dinh la didm A 'l . b . ; ed true ddi xiing la dudng thang x =-—', hudng be 16m /2fl' 4a, booktoan.com len tren khi o > 0 va xudng dudi khi o < 0. 4
' A.__
28
Pheptinhtiendothj Cho ham sd y =f{x) cd d6 thi (G) ; p va ^ la hai sd khdng am. • Khi tinh tie'n (G) len tren q don vi, ta dupe d6 thi cua ham sd y -f{x) + q. • Khi tinh tie'n (G) xudng dudi q don vi, ta dupe dd thi ciia ham sd y =^x) - q. • Khi tinh tie'n (G) sang trai p dan vi, ta dupe dd thi ciia ham sd y ~f{x + p). • Khi tjnh tie'n (G) sang phai p don vi, ta dupe dd thj ciia ham sd y -fix - p).
B. DE BAI §1. DAICl/ONGVfiHAMSd Khai niem ham so 2.1. Dudng tron tam O ban kfnh r khdng phai la d6 thi ciia mdt ham sd. Nhung ntfa dudng tron g6m cae didm ed lung dd khdng am ciia dudng tron tam 0 ban kfnh r (h. 2.1) la dd thi eiia mdt ham sd. Hay vi^t bidu thiic xac dinh ham sd dd va cho bie't tap xac dinh cua nd, bie't rang dudng tron tam O ban kfnh r \a tap hpp cic didm cd toa d6 2
2
2
(x ; y) thoa man he thu'c x + y = r 2.2. Tim tap xac dinh cua cdc ham sd sau : a)y =
Hinh 2.1 Nita dir6ng tron ban kfnh r = 2
x-1 x^-l V2X+T 2x^ - X - 1
c)y =
3x + 4 (X - 2)Vx + 4 '
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2.3. Cho ham sd ne'u X > 0,
x+1
f{x) =
}[xV\ I x-1
ne'u - 1 < X < 0 .
a) Tim tap xac dinh cua ham sdy =/(x). b)Tfnh/(0),/(2),/(-3),/(-l). 2.4. Cho ham sd
fix)
=x^+ylx~l.
a) Tim tap xac dinh cua ham sd. b) Dung bang sd hoac may tfnh bo tiii, tfnh gia tri gdn dung eiia /(4), fi-Ji) .fin) chinh xac de'n hang ph^n tram. Sir bien thien cua ham so
2.5. Hay lap bang bie'n thien ciia ham sd cd dd thi la niia dudng tron cho tren hinh 2.1. 2.6. Dd thi eiia mdt ham sd xac dinh tren R dupe cho tren hinh 2.2. Dua vao dd thi, hay lap bang bie'n thien cua ham sd dd. Hay cho bie't gia tri Idn nha't hay nho nha't ciia ham sd (ne'u cd). 2.7. Bang each xet ti sd Z^^zWXfi)^ j^-y X2 - X i
neu su bie'n thien eiia cac ham sd sau (khdng yeu e^u lap bang bie'n thien cua nd) tren cac khoang da cho : a) y = x + 4x + 1 tren mdi khoang (-oo ; -2) va (-2 ; +oo); b) y = -x^ + 2x + 5 tren mdi khoang (-oo ; 1) va (1 ; +oo); e) y = d) y = 30
X
x +1
tren mdi khoang (-oo ; -1) va (-1 ; +co);
2x + 3 tren mdi khoang (-oo ; 2) va (2 ; +co). booktoan.com -x + 2
Ham so' c h i n va ham so le
2.8. Cd hay khdng mOt ham sd xac dinh tren E viia la ham sd ehSn vira la ham sd le ? 2.9. Cho hai ham sd y = fix) vay = g(x) xac dinh tren K. Dat Six) = fix) + g(x) vaP(x) =/(x)^(x). Chiing minh rang : a) Ne'u y = fix) vay = g{x) la nhflng ham sd chan thi y = S{x) va y = Pix) Cling la nhiing ham sd chan. b) Ne'u y =fix) va y = gix) la nhiing ham sd le thi y = Six) la ham sd le va y = Pix) la ham sd chan, c) Ne'u y = fix) la ham sd chan, y = gix) la ham sd sd le thi y = Pix) la ham sd le. 2.10. Xet tfnh chan le ciia eac ham sd sau : a) y = 3x'* + 3x^ - 2 ;
b) y = 2x^ - 5x ;
e) y = X tx| ;
d) y = Vl + x + Vl - x ;
e) y = V1 + X - v l - X. Tjnh tie'n do thj song song vdi true toa do
2.11. Trong mat phing toa dd, cho cae didm A(-l ; 3), Bi 2 ; -5), da ; b). Hay tinh toa dd cae didm ed dupe khi tinh tid^n cac didm da cho : a) Len tren 5 don vi;
b) Xudng dudi 3 don vi ;
c) Sang phai 1 don vi;
d) Sang trai 4 don vi.
2.12. Ham sd y = 4x - 3 ed dd thi la dudng thing id). a) Gpi idi) la dudng thing ed dupe khi tinh tie'n id) len tren 4 don vi. Hoi (rfj) la dd thi ciia ham sd nao ? b) Gpi id2) la dudng thing cd dupe khi tinh tid^n id) sang trai 1 don vi. Hoi id2) la d6 thi cua ham sd nao ? e) Em cd nhan xet gi vd hai k^t qua tren ?
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_2 2.13. Gia sii ham sd y = — cd dd thj la (//). a) Ne'u tjnh tie'n (//) xudng dudi 3 don vi thi ta dupe d6 thi cua ham sd nao ? b) Ne'u tinh tie'n (//) sang phai 2 don vi thi ta dupe dd thi ciia ham sd nao ? e) Ne'u tinh tie'n (//) len tren 1 don vi r6i sang trai 4 don vi thi ta dupe dd thi eua ham sd nao ?
§2. HAM S6 BAC N H X T 2.14. Ve dd thj ciia mdi ham sd sau :
1 a)y = 2x-3;
J3
b) y = - - x +1 ;
c) y = — x + 2.
2.15. Trong mdi trudng hpp sau, tim eac gia tri ciia k sao cho dd thi ciia ham sd y = -2x + kix+\) a) Di qua gd'c toa dd O ; b) Di qua didm M(-2 ; 3 ) ; c) Song song vdi dudng thing
y-yj2x.
2.16. Tim cac cap dudng thing song song trong cae dudng thing eho sau day : a) 3y - 6x + 1 = 0 ;
b) y = -0,5 x - 4 ;
c)y = 3 + | ;
d)2y + x = 6 ;
e) 2 x - y = 1 ; f)y = 0,5x+ 1. 2.17. Ve dd thi cua mdi ham sd sau vk lap bang bie'n thien ciia nd : a) y = I 3x + 5| ; b)y = - 2 | x - l | ;
c)y = - i i 2 x + 3 | + | . 2.18. Trong mdi trudng hpp sau, xac dinh avkb sao cho dudng thing y = ax + b a) Cat dudng thing y = 2x + 5 tai didm cd hoanh dd bing -2 va cat dudng thing y = ~3x + 4 tai didm cd lung dd bang -2 ;
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32
b) Song song vdi dudng thing y= - x v^ di qua giao didm cua hai dudng thing y = - x ^ + 1 vay = 3x + 5. 2.19. a) Cho didm Aix^ ; y^. Hay xac dinh toa dp ciia didm B, bie't ring B dd'i xu^g vdi A qua true hoanh. b) Cbiing minh ring hai dudng thing y = x - 2 va y = 2 - x ddi xiing vdi nhau qua true hoanh. c) Tim bidu thiic xac dinh ham sd y = fix), biet ring d6 thi eiia nd la dudng thing dd'i xiing vdi dudng thing y = -2x + 3 qua true hoanh. 2.20. a) Cho didm AixQ ; yo). Hay xac dinh toa dd cua didm B, bie't ring B ddi xiing vdi A qua true tung. b) Chiing minh ring hai dudng thing y = 3x + 1 va y = -3x + 1 ddi xiing vdi nhau qua true tung. c) Tim bidu thiic xac dinh ham sd y = fix), bie't ring dd thi cua nd la dudfng thing ddi xiing vdi dudng thing y = 0,5x - 2 qua true tung. 2.21. Mdt tia sang ehie'u xien mot gde 45° de'n didm O tren bd mat eua mdt cha't long thi bi khue xa nhu hinh 2.3. Ta lap he toa dp Oxy nhu da thd hien tren hinh ve. a) Hay tim ham sd y - fix) ed dd thi triing vdi dudng di eiia tia sang ndi tren. b) Lap bang bie'n thien ciia ham
s6y-fix).
2.22. a) Tim didm A sao cho dudng thing y == 2mx + 1 - m ludn di qua A, du m la'y ba't cii gia tri nao. b) Tim didm B sao eho dudng thing y = mx ^ 3 - x ludn di qua B, dii m la'y ba't cii gia tri nao. 2.23. Trong mdi trudng hpp sau, tim eac gia tri ciia m sao eho a) Ba dudng thing y = 2 x , y = - 3 - x v a y = mx + 5 phan biet va ddng quy. b) Ba dudng thing y = -5(x + l ) , y = mx + 3va y = 3x + m phan biet va ddng quy.
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33
§3.HAMS6BACHAI 2 2
2.24. Cho ham s6 y = - x a) Khao sat su bie'n thien va ve d6 thi iP) ciia ham sd da eho. b) Ne'u tinh tie'n iP) len tren 2 don vi thi ta dupe dd thi ciia ham sd nao ? c) Ne'u tinh tie'n iP) xudng dudi 3 don vi thi ta dupe dd thi eiia ham sd nao ? yl3 2 2-25. Cho ham sd y = ——x a) Khao sat su bien thien va ve dd thi iP) eiia ham sd da cho. b) Ne'u tinh tien (f) sang phai 1,5 don vi thi ta dupe dd thi eiia ham sd nao ? c) Ne'u tinh tie'n (/*) sang trai 2 don vi thi ta dupe dd thi eua ham sd nao ? 2
2.26. Cho ham sd y = 2x cd dd thi la parabol (f). Phai tinh tie'n (/*) nhu thd nao de dupe dd thi eua ham sd a) y = 2x^ + 7 ;
b) y = 2x^ - 5 ;
c)y = 2(x + 3)^ ;
d)y = 2 ( x - 4 ) ^
e) y = 2(x - 2)^ + 5 ; /) y = 2x^ - 6x +1 ? 2.27. Khong ve dd thi, tim toa dp dinh, phuong trinh true ddi xiitig eiia mdi parabol sau day. Tim gia tri nhd nha't hay Idn nha't eiia mdi ham sd tuong ling. a)y = 2(x + 3 ) ^ - 5 ; b)y = - ( 2 x - l ) ^ + 4 ; e)y = - V2x^ + 4x 2.28. Khao sat su bie'n thien va ve dd thi ciia cae ham sd sau : a) y = x^ + X + 1 ;
b) y = -2x^ + x - 2 ;
c) y = -3? + 2x -1 ;
d) y = i x ^ - X + 2.
2.29. Cho ham sdy = -x + 4x - 3. a) Khao sat su bie'n thien va ve dd thi ciia ham sd da cho. b) Dua vao dd thi, hay neu cac khoang tren dd ham sd chi nhan gia tri duong. c) Dua vao dd thi, hay neubooktoan.com cac khoang tren dd ham sd chi nhan gia tri am. 34
a-BTDsio.Nc-e
2.30. Cung yeu cdu nhu bai 2.29 dd'i vdi cac ham sd sau : ^ 2 3 a)y = x - X + - ; b) y - -2x^ + 3x - -^ ; c) >- = 0,5x'' - 3x. 2.31, Ve do thi ciia mdi ham sd sau rdi lap bang bien thien cua nd : a) y = ^x^ + 2x - 6 b) y = |-0,5x' + 3x-2,5 |. 2 2.32. Ve do thi ciia mdi ham sd sau rdi lap bang bie'n thien eua nd : y!,t^
a) fix)0= b)/(x) =
f-2x + 1
.
.
ne'u x> 0
[x^ + 4x + 1 ne'u X < 0 ; -x^-2
ne'u X < 1
2x - 2x - 3 ne'u x > 1. 2.33 , Ve dd thi ciia ham sd y = -x + 5x + 6. Hay sir dung dd thi dd bien luan
theo tham sd m sd didm ehung eua parabol y - -x + 5x + 6 va dudng thing y = m. 2.34 , Mdt parabo! cd dinh la didm /(-2 ; -2) va di qua gd'c toa dp.
a) Hay eho bie't phuong trinh true ddi xutig cua parabol, bie't ring nd song song vdi true tung. b) Tim didm ddi xiing vdi gde toa dp qua true ddi xiing trong eau a). c) Tim ham sd cd dd thi la parabol da eho. 2.35 a) Kf hieu (P) la parabol y = ax + bx + c ia ^ 0). Chiing minh ring ne'u
mdt dudng thing song song vdi true hoanh, cat iP) tai hai didm phan biet A va B thi trung didm C ciia doan thing AB thude true dd'i xiing ciia parabol (P). b) Mpt dudng thing song song vdi true hoanh cat dd thi (P) ciia mdt ham sd bac hai tai hai didm M(-3 ; 3) va A'(l ; 3). Hay cho bie't phuong trinh true ddi xiing eua parabol (P).
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35
•J
2.36. Ham sd bac hai fix) -ax
1
+ bx + c c6 gia tri nho nh^t bang — khi x = ^^
va nhan gia tri bang 1 khi x = 1. a) Xac dinh eac he sd a, b va c. Khao sat su bie'n thien va ve dd thi (?) eiia ham sd nhan dupe. b) Xet dudng thing y = mx, kf hieu bdi id). Khi id) cit (?) tai hai didm A va B phan biet, hay xac dinh toa dd trung didm cua doan thing AB.
BAI TAP 6 N TAP CHl/ONG II 2.37. Chiing minh ring y = 0 la ham sd duy nha't x^c dinh tren R va cd d6 thi nhan true hoanh lam true ddi xiing. Hudng din. Til dinh nghia ham sd ta ed nhan x6t ring mdi dudng thing song song vdi true tung thi cat dd thi ciia ham sd tai khdng qua mdt didm. 2.38. Gia sir y =fix) la ham sd xac dinh tren tap dd'i xiing 5 (nghia la ne'u x G 5 thi -X e S). Chiing minh ring : a) Ham sd Fix) = - |/(x) +/(-x)] la ham sd chSn xdc dinh tren 5. b) Ham sd Gix) = i |/(x) -/(-x)] la ham sd le xac dinh tren 5. 2.39. Gpi A va S la hai didm thude dd thi ciia ham sd/(x) = (m - l)x + 2 va cd hoanh dd lin lupt la - 1 va 3. a) Xac dinh toa dd ciia hai didm A va B. b) Vdi didu kien nao ciia m thi didm A nim d phfa tren true hoanh ? e) Vdi didu kien nao cua m thi didm B nim d phfa tren true hoanh ? d) Vdi didu kien nao eiia m thi hai didm A va S ciing nim d phfa tren true hoanh ? Tii dd hay tra ldi cau hoi ; Vdi didu kien nao eiia m thi fix) > 0 vdi mpi X thude doan [-1 ; 3] ? 2.40. Cho ham sd y = -3x^ cd dd thi la parabol (?). a) Ne'u tinh tie'n (?) sang phai 1 don vi rdi tinh tie'n parabol viira nhan dupe xudng dudi 3 don vi thi ta dupe dd thi eiia ham sd nao ? b) Ne'u tinh tie'n (?) sang trai 2 don vi rdi tinh tie'n parabol viira nhan dupe len tren 2 don vi thi ta dupe d6 thi eiia ham sd nao ? booktoan.com 36
2.41. Tim ham sd bac hai cd dd thi la parabol (?), bidt rang dudng thing y = -2,5 cd mdt didm ehung duy nha't vdi (?) va dudng thing y = 2 cit (?) tai hai didm cd hoanh dd la -1 va 5. Ve parabol (?) ciing cac dudng thing y = -2,5 va y = 2 tren ciing mdt mat phing toa dd.
Gidl THifiU M C T
S 6 CAU HOI TRAC
NGHlfeM K H A C H QUAN
Trong cdc bdi tO: 2.42 de'n 2.49, hay chon phuang an tra Icfi dung trong cdc phuang dn dd cho. 2.42. Tim didm thude dd thi eiia ham sd y = - x - 2 trong eac didm cd toa dp la (A) ( 1 5 ; - 7 ) ;
(B) (66 ; 20) ;
(C)(V2-1;V3);
(D) (3 ; 1).
2.43. Ham sd cd dd thi triing vdi dudng thing y = x + 1 la ham sd
(A)y=(VxTl) ;
(B)y=i—-^;
(C) y = x(x + 1) - x^ + 1 ;
(D) y - ^^^^^^
2.44. Dudng thing song song vdi dudfng thing y= V2 x la (A) y = 1 - V 2 x ;
(B) y - -|=x - 3 ;
(C)y+V2x=2;
( D ) y - - | . x = 5.
2.45. Mudn cd parabol y = 2(x + 3) , ta tinh tie'n parabol y = 2x (A) Sang trai 3 don vi; (B) Sang phai 3 don vi ; (C) Len tren 3 don vi ; (D) Xudng dudi 3 don vi. 2
2
2.46. Mudn ed parabol y = 2(x + 3) - 1 , ta tinh tie'n parabol y = 2x (A) Sang trai 3 don vi r6i sang phai 1 don vi ; (B) Sang phai 3 don vi rdi xudng dudi 1 don vi; (C) Len tren 1 don vj rdi sang phai 3 don vi ; (D) Xudng dudi 1 don vi rdibooktoan.com sang trai 3 don vi. 37
2.47. True ddi xiing eiia parabol y = -2x + 5x + 3 la dudng thing (A)x=- ;
(B)x = - -
(C)x=- ;
(D)x = - -
2.48. Ham sd y = 2x^ + 4x ~ 1 (A) Dong bie'n tren khoang (-^ ; -2) va nghich bie'n uen khoang (-2 ; + oo); (B) Nghich bie'n tren khoang (-oo ; -2) va ddng bie'n tren khoang (-2 ; + oo); (C) Ddng bie'n tren khoang (-co ; -1) va nghich bie'n tren khoang (-1 ; + co); (D) Nghich bie'n tren khoang (-oo ; -1) va ddng bie'n tren khoang (-1 ; + co). 2.49. Ham sd y - -x - 3x + 5 cd 3 (A) Gia tri Idn nha't khi x = - ;
(B) Gia tri Idn nha't khi x = - -
(C) Gia tri nho nha't khix= - \
(D) Gia tri nho nha't khi x = - • -
Trong mdi bdi tif bdi 2.50 den bdi 2.52, hay ghep mdi thanh phan cua cdt trdi vai mot thanh phan thich hap d cot phai deduac khdng dinh dung. 2.50. a) Didm ( 2 ; 2) la dinh ciia parabol b) Didm
r 1
P
I 2'2,
la dinh cua parabol
. l)y = 2x' + 2x+ 1. 2)y = x ' - x + l . 3)y = -0,25x' + x + 1.
2.51. Xet parabol {P) : y = ax^ + bx + c a) Chic chin (?) cd dinh nim d phfa dudi 1) ne'u a 0 va c < 0 b) Chic chin (?) ed dinh nim d phfa tren 3) ne'u <3 < 0 va c > 0 true hoanh 4) ne'u (3 > 0 va t > 0
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38
2.52. Xet parabol (?) : j = ax" + /jx + c vdi a < 0, t^ = \? ~ 4ac. a) Chic chin (?) cit true hoanh tai hai didm ed hoanh dp duong
l)ne'u A > 0 , & < O v a c < 0 2) ne'u A > 0, 6 > 0 va (•> 0
b) Chic chin (?) cit true hoanh tai hai 3) neu A > 0 , ^ < 0 v a c > 0 didm ed hoanh dd am 4) ne'u A > 0 , & > O v a c < 0
C. DAP SO - HUONG DAN - LOi GIAI 2.1. y=ylr
~ x^ , xac dinh tren doan [-/•; r].
Chu y. Ham sd y = -Vr^ - x"^ cd dd thi la nia dudng tron gdm eac didm thupc dudng tron dang xet va ed tung dp khong duong (cGng ed tap xac djnh la [~r ; r]). 2.2. a ) R \ { - l ; 1}. b)(-|;+oo)\{l). e)(-4;+0D)\{2}.
2.3. a) [-1 ; +00).
b)/(0) = -1 ; /(2) = I
; / ( - l ) = 0 ; /(-3) khdng xac dinh.
2.4. a) [1 ;+«)).
b)/(4)= 16+ V3 =« 17,73; / ( 7 2 ) ^ 2 , 6 4 ; fin) ^ 11,33. 2.5. X
-2
0 ^ _ ^
y=
yJ4-X^
0
-^-"^
2
2 ___^ ^""^"•^—-^ 0
booktoan.com 39
2.6. -1
—00
+00
2,4
+C0
+00
1,6
-4,4
Ham sd cd gia tri nhd nha't bing -4,4 khi x = 2, nhung khdng cd gia tri Idn nha't.
2.7. , ) / ( ^ 2 ) - / U | ) , ^ ^ ^ , ^ ^ 4 . Xj
Xj
Tren khoang (-oo ; - 2 ) , ta cd X2 + x^ + 4 < 0 nen him sd nghich bie'n. Tren khoang (-2 ; +GO), ta ed X2 + XJ + 4 > 0 nen ham sd ddng bid^n.
X2 - Xi
Tren khoang (-co ; 1), ta cd -X2 - Xj + 2 > 0 nen ham sd ddng bie'n. Tren khoang (1 ; +co), ta cd -X2 - Xj + 2 < 0 nen ham sd nghich bie'n. c) Vdi hai sd phan biet Xj va X2 thude tap xac dinh eua ham sd', ta cd :
fiX2)-fix,) =
Xf
X2 + 1
Xi + 1 (Xi + 1)(X2 + 1) '
/(X2)-/(X^)^
X2 - Xj
Xi
1
(Xi + 1)(X2 + 1) '
Dodo: - Neu Xi < - 1 va X2 < - I thi (X] + 1)(X2 + 1) > 0 va
1
(Xi + 1)(X2 + 1) >o,
suy ra ham sd ddng bie'n tren khoang (-co ; - 1 ) . - Ne'u xi > - 1 va X2 > - 1 thi (xj + 1)(X2 + 1) > 0 va suy ra ham sd cung ddng bien tren khoang (-1 ; +00).
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40
1 (Xi + 1)(X2 + 1)
>0,
d) -^^^2) / U i ) ^ Z . Tiif dd suy ra ham sd da eho X2 - X| (-X2 + 2)(-Xi + 2) ddng bie'n tren mdi khoang (-00 ; 2) va (2 ; +co). 2.8. De tha'y ham sd' y = 0 la ham sd xac dinh tren R, viia la ham sd chan, viia la ham sd le. Gia sir ham sd y =/(x) la mot ham sd bat ki ed tfnh eha't nhu the'. Khi dd vdi moi x thude M, ta cd : fi-x) -fix)
( v i / i a ham sd chan);
/(-;c) = -/(x) ( v i / l a h a m s d l e ) . TCr dd suy ra vdi moi x thude M, xay rafix) = -fix), nghia \afix) = 0. Vay y = 0 la ham sd duy nha't xac dinh tren R, vixa la ham sd chin, viia la ham sd le. 2.9. a) De dang suy ra tir gia thie't va dinh nghia ham sd chan. b) Vdi X tuy y thude M, ta cd : /(~x) = -fix) va gi-x) = -g(x) (vi / va g la nhitng ham sd l e ) ; do dd 5(-x) - / ( - x ) + gi-x) = -fix) - gix) = -\fix) + gix)] = -Six), ?(-x) =/(-x)^(-x) = [-fix)][-gix)]
=fix)gix) = ?(x).
vay y = iS(x) la ham sd le va y = ?(x) la ham sd chin. c) Vdi X tuy y thuoc E, ta cd :/(-x) =/(x) va gi-x) = -gix)
( v i / l a ham sd
chin va g la ham sd l e ) ; do dd ? ( - x ) =fi-x)gi-x)
=/(x)[-^(x)] = ~fix)gix) = -Pix).
Vay y = ?(x) la ham sd le. 2.10. a) Ham sd chan (tdng ciia ba ham sd chin). b) Ham sd le (long ciia hai ham sd le). c) Ham sd le (tich cua ham sd le y = x va ham sd chan y = |x|). d) Tap xac dinh eua ham sd/(x) = Vl + x + Vl - x la doan [-1 ; 1]. Vdi m o i x thude doan [-1 ; 1], ta ed :
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41
fi-x) = Vl - X + Vl + x =/(x). vay y =/(x) la ham sd chin. e) Tap xac dinh eda ham sd gix) = Vl + x - Vl - x la doan [-1 ; 1]. Vdi moi X thude doan [-1 ; 1], ta ed :
^(-x) = V T ^ - Vl + x = -gix). vay y = gix) la ham sd le. 2.11. a) Khi tinh tien len tren 5 don vi, ta dugfe : Ai-l ; 3) h^ A^i-\ ; 8); B(2 ; -5) h^ ?i(2 ; 0) ; da ; b) \-^ Cyia ;b + 5). b) Khi tinh tie'n xudng dudi 3 don vi, ta dugfe : A(-l ; 3) h^ ^2(^1 ; 0 ) ; ^(2 ; -5) h^ ?2(2 ; -8) ; Cia ; b) h^ C2{a •,b-3). c) Khi tinh tidn sang phai 1 don vi, ta duoc : A(-l ; 3) h^ A^iO ; 3); 5(2 ; -5) h^ ^3(3 ; -5) ; da ; b) h^ 03(0 + 1 ; &). d) Khi tinh tie'n sang trai 4 don vi, ta duoc : Ai-l;
3) h^ A4(-5 ; 3); B(2 ; - 5 ) h^ S4(-2; -5) ; C(a ; b) \-^
C^ia-4\b).
2.12. a) (Ji) la dd thi eiia ham sdy = ( 4 x - 3) + 4 hay y = 4 x + 1. b) (^2) '^ ^^ thi eiia ham sd y = 4(x + 1) - 3 hay y = 4x + 1. e) Dudng thing y = 4x + 1 cd thd ed duoc bing each tjnh tie'n dudng thing y - 4x ~ 3 theo hai each nhu trong a) va b). -2 2.13. a) Tinh tie'n (//) xudng dudi 3 don vi, ta duoe dd thi ham sd y =
3,
-3x - 2 hayy=—-— b) Tinh tie'n (//) sang phai 2 don vi, ta duoc dd thi ham sd y =
-2 x-2
c) Tinh tie'n (//) len tren 1 don vi rdi sang trai 4 don vi, ta duoc do thi . . . . -2 _ . x+2 + 1, hay y 7 ham so y = ^ x+4 -^ x + 4
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42
2.14. a), b) Hpc sinh tu giai. c) D6 thi hinh 2.4. Giao didm vdi true tung : (0 ; 2). Giao didm vdi true hoanh :
2A5.a)k
= 0.
b ) ^ = 1.
C)k^2+yl2 2.16. cac cap dudng thing song song la a) va e) ; b) va d ) ; c) va/).
Hinh 2.4
2.17. a) D6 thi hinh 2.5.a).
c)
a)
r-2x + 2 khi X > 1 DO thi hinh 2.5.b). b) Ham sd cd the vie't dang y=< 2x - 2 khi X < 1 - x + 1 khi X > -— e) Ham sd ed the vie't dang y = •
. Dd thi hinh 2.5.c).
x + 4 k h i x < -— 2 (Hpc sinh tu lap bang bie'n thien). 2.18. a) Tren dudng thing y = 2x + 5, didm ed hoanh do bing - 2 la Ai-2 ; 1). Tren dudng thing y ^ -3x + 4, diem cd tung dp bing - 2 la 5(2 ; -2). Vay dudng thing cin tim di qua hai didm booktoan.com A va B. Tit dd, a va b phai thoa man he 43
f~2a +fo= 1 [2a + b=^ -2. <. 3 , 1 S u y r a a - - - ,b = --h) Giao didm M eiia hai dudng thing y = - —x + 1 va y = 3x + 5 cd toa dd la nghiem ciia he phuong trinh
1 y=--x+l y = 3x + 5.
He nay ed nghiem (x ; y) = - - ; — ]. Vay dudng thing cin tim song 1
/' 8 11 ^
song vdi dudng thing y = - x va di qua didm Ml - - ; — I. Tut dd suy ra 1 2
. ,
15 7
a = — \ab -—2.19. a) B(xo ; -yo). b) Mudn chiing minh hai dudng thing (c/,) va ((^2) dd'i xiing vdi nhau qua true hoanh, ta ehiing minh ring ne'u A(xo ; yo) la mdt didm tuy y thude idi) thi didm ddi xiing vdi A qua true hoanh, tiic la diem BixQ ; -yo) thude (J2) va ngupe lai. That vay, gpi id^) la dudng thing y = x - 2, ((^2) la dudng thing y - 2 - x, ta cd ^(^0 ; yo) e (^1) '^yo = xo-2 <^ -yo = 2 - xo <^ ?(xo ; -yo) e id2). TCr dd suy ra dpem. e) Tuong tu nhu cau tren, ta dd dang chiing minh dupe ring dd thi eiia hai ham sd y =/(x) va y = -fix) ddi xiing vdi nhau qua true hoanh. Do dd, dudng thing ddi xung vdi dudng thing y = -2x + 3 qua true hoanh la dd thi eiia ham sd y ~ -(-2x + 3), tde la ham sd y = 2x - 3. 2.20.a)?(-Xo;yo). b) Chiing minh tuong tubal 2.19.b). c) y ^ -0,5x -2. Gai y. Trudc het chdng minh ring dd thi ciia hai ham sd y =fix) va y =/(-x) ddi xiing vdi nhau qua true tung. booktoan.com 44
-^11
\ J-, ^
\~x
khi X < 0
2.21. a)/(x)='^ [-2x
khi X > 0.
b) Hpe sinh tu lap bang bie'n thien. 2.22. a) Gia sii didm A cin thing y = 2mx + 1 - m yo = 2w XQ + xay ra vdi mpi m. Didu
tim ed toa d6 (XQ ; yo). Khi dd, vi A thude dudng vdi mpi m nen ding thu'c 1 - m, hay (2 Xo - l)m + 1 - yo = 0 dd ehi cd thd xay ra khi ta cd ddng thdi 2xo - 1 = 0
va 1 - yo = 0, nghia la XQ = — va yo = 1. Vay toa dd eiia ^ la (—; 1). Ngupe lai, de thiy gia tri ciia ham sdy = 2mx + 1 - m tai x = —ludn bing 1 vdi mpi m, chiing to dd thi eiia nd ludn di qua didm Ai—; 1) vdi mpi m. b) B(0 ; - 3 ) . Goi y. Cach giai tuong tu eau a). 2.23. a) Hai dudng thing y = 2x va y = - 3 - x cit nhau tai M ( - l ; -2). Dudng thing thd" ba y = mx + 5 cung di qua didm M khi va chi khi - 2 = m(-l) + 5, tiic la m = 7. Thii lai ta tha^y m thoa man didu kien cua diu bai. b) Hai dudng thing y = -5(x + 1) va y = 3x + m cit nhau tai N
^ m + 5 5m - 15
Dudng thing y = mx + 3 cung di qua A^ khi va chi khi 5m - 15 f •=m\ 8
m + 5^ —
^
+ 3.
Giai phuong trinh tren ddi vdi in m, ta dupe m = - 1 3 v a m = 3. -V6im = - 1 3 , ba dudng thing y = -5(x + 1), y = -13x + 3 va y = 3x - 13 ddng quy lai didm N^H ; -10). - Vdi m = 3, hai dudng thing y = mx + 3 va y = 3x + m trung nhau va triing vdi dudng thing y = 3x + 3. Do dd trudng hpp nay bi loai. Ket ludn. m = - 1 3 . 2.24. a) Hpe sinh tu giai.
b) y = - x ^ + 2.
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c) y = - x ^ - 3. 45
2.25. a) Hpc sinh tu giai.
b)y- - ^ ( x - 1 , 5 ) '
c)y = - ^ i x + 2f 2.26. a) Tinh tie'n (?) len tren 7 don vi. b) Tinh tie'n (?) xudng dudi 5 don vi. c) Tinh tie'n sang trai 3 don vi. d) Tinh tie'n sang phai 4 don vi. e) Tinh tie'n sang phai 2 don vi rdi tinh tie'n tie'p len tren 5 don vi. /) Tjnh tie'n sang phai 1,5 don vi rdi tinh tie'n tie'p xudng dudi 3,5 don vi. 2.27. Ke't qua dupe neu trong bang sau Dinh
Parabol
(-3;-5)
y = 2(x + 3)^-5
4; 4)
y = -(2x-l)^ + 4 = -4(x-^)^ + 4 y = -^^x^ + 4x= -V2(x - V2)^ + 2%^
True Gia tri Gia tri ddi nhd Idn xiing nhat nhit x = -3
-5
1
( V 2 ; 2 ^ ) x=V2
4 2V2
2.28. a) Ta cd the vie't ham sd y = x + x + 1 dudi dang y=\^+\
3 ^-4'
Ttr dd suy ra dd thi ciia nd la mpt parabol hudng bd iQm len tren va ed dinh tai
i 1 ; ham sd da eho nghich bie'n 2'4
tren khoang (-c» ; - - ) , ddng bie'n tren
Hinh 2.6
4t
khoang (-—; +QO) va cd gia tri nho nha't bing — khi x = -— Hpc sinh tu lap bang bie'n thien. 46
booktoan.com
Dd ve dd thi eiia ham sd nay, ta lap bang mdt vai gia tri ciia nd nhu sau X
-2
-1
y
3
1
1 2 3 4
0
1
1
3
Dd thi ciia ham sd ed dang nhu hinh 2.6. b) Dua ham sd da cho vd dang y = - 2
15 ^"4
TiJf dd suy ra ham sd
ddng bie'n tren khoang (-oo ; —), nghich bie'n tren khoang ( —; +oo) va cd gia tri Idn nhit bing —— khi x =—. Hpc sinh tu lap bang bien thien va a 4 ' ve dd thi ciia nd. c) Hpc sinh tU giai. d) Hpc sinh tu giai. 2.29. a) Ham sd y = -x^ + 4x -3 cd thd vie't dupe dudi dang y = -(x-2)2+l. Tii do suy ra ham sd ddng bie'n tren khoang (-co ; 2), nghich bie'n tren khoang (2 ; +oo). Bang bie'n thien : '00
+00
-00
—00
y= -X + 4x - 3
Ham sd cd gia tri Idn nha't bing 1 khi x = 2. Q6 thi eua nd la mdt parabol di qua eac didm (0 ; - 3 ) , (1 ; 0), (2 ; 1), (3 ; 0), (4 ; - 3 ) (h.2.7). Tir dd thi ta tha'y : b) Ham sd ehi nhan gia tri duong ne'u x e (1 ; 3). c) Ham sd ehi nhan gia tri am ndu X e (-00 ; 1) u ( 3 ; +oo).
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Hinh 2.7
47
2.30. a) Hpc sinh tu khao sat su bien thien cua ham sd. Ham sd cd dd thi nhu hinh 2.8a. Ham sd nhan gia tri duong vdi mpi x e R. b) Hpe sinh tu khao sat su bie'n thien cua ham sd. Ham sd cd dd thi nhu hinh 2.8b. 3 3 Ham sd nhan gia tri am vdi mpi x ^— (khi x = — . ham sd nhan gia tri bing 0). c) Hpe sinh tu khao sat su bie'n thien eiia ham sd. Ham sd ed dd thi nhu hinh 2.8e.
b)
a)
c)
Hinh 2.8
Ham sd nhan gia tri am ndu x e (0 ; 6) va nhan gia tri duong ndu X G (-00 ; 0) LJ (6 ; +oo).
2.31. a) y = - x ^ + 2 x - 6 Q6 thi (h. 2.9a). Bang bie'n thien —00
-6
-2
+00
,+00
booktoan.com 48
+00
b)y = |-0,5x^ + 3 x - 2 , 5 | Dd thi (h. 2.9b). y \ \
•2,5
.2
\ \
0 -2
/
/
' ^\ ' \
A
h
/ -> // \ ^
1^ 1'
/
I
/
•..
1
/
/
X • ' ^
^ — ' '
'' /
\ X \\ \
\
1
a)
b) Hinh 2.9
Bang bie'n thien
2.32. a) y =
—00
+00
+00
+00
-2x + 1
ne'u x> 0
x^ + 4x + 1 ndu X < 0.
Dd thi (h. 2.10a). Hpc sinh tu lap bang bien thien. b)y =
-x^-2
ne'u X < 1
2x^ - 2 x - 3
ne'u x > I.
Dd thi (h. 2.10b). Hpc sinh tu lap bang bie'n thien.
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4-BTOSlO.NC-A
49
b)
a) Hinh 2.10
2.33. Hoc sinh tu ve dd thi. Do parabol hudng bd 16m xudng dudi va ed dinh tai didm p^" ^ i 2 - j nen :
- Ndu m> 12— thi dudng thing va parabol khdng cd diem ehung. - Neu m= 12— thi dudng thing va parabol cd mot diem ehung. - Neu m < 12— thi dudng thing va parabol cd hai diem ehung phan biet. 2.34. a) Phuong trinh true ddi xiing la x = - 2 . b) Diem ddi xdmg vdi 0 ( 0 ; 0) qua true x = - 2 la diem M(-4 ; 0). 2
,
c) Ta phai tim a (a ^ 0), b va. c sao cho ham s6 y = ax + bx + c c6 do thi la parabol dinh /(-2 ; -2) va di qua didm O. Tii gia thie't ta ed cac he thde sau : „ A b^ -4ac ^ . ^ b - - - = -2 ; - - — = = - 2 va c = 0. 2a 4a 4a 1 Tir dd tinh dupe a=50
1 2 ,b = 2,c = 0va ham sd can tim la y = - x + 2x.
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4-BTDSlO.NC-B
9
1
2.35, a) Ta da biet true ddi xuTig ciia parabol y = ax + ^x + c la dudng thang __b_ 2^' Gia su (d) la dudng thing da cho (song song vdi true hoanh). Ta bie't ring id) la dd thi cua ham sd khong ddi y = m vdi m la mot sd nao dd. Gia thie't cho id) eit (?) tai hai didm phan biet A va B c6 nghia la phuong trinh ax
+ bx + c = m hay ax+bx
+ c-m
=0
(1)
cd hai nghiem phan biet ; hon nUa, hai nghiem a'y chinh la cac hoanh dp x^ ciia didm A va xg eua diem B. Theo dinh li Vi-et, ta cd x^ + Xg = — Do dd trung diem C ciia doan thine AB ed hoanh do la Xr = — •
^
•
^
—= -— 2
2a
Didu dd ehung td diem C thuoc dudng thing x = - - — , tiJc la thude true 2a ddi xiing cua parabol (?). Chuy. Dudng thing id) song song vdi true hoanh nen vudng gde vdi true ddi xutig ciia (?). Do dd, khi id) cit (?) tai hai diem AvaB thi hai diem a'y ddi xiing vdi nhau qua true ddi xiing va trung diem C cua doan AB phai ihudc true ddi xiing. b) Ap dung ke't qua tren, trung diem K eua doan MN phai thude true ddi xiing ciia parabol (?). Diem K cd hoanh dd la ——— = - I . Vay true ddi xiing ciia parabol (?) cd phuong trinh la x = - 1 . 2.36. a) • Vl ham sd cd gia tri nho rtha't b i n s — khi x = — nen --— = — va 4 2 2a 2 A b'^ - 4ac 3 ^ —- suy raa = -b va -a-\-4c = 3. 4a 4a 4 Vi ham sd ed gia tri bing 1 khi x - 1 nen/(I) = a + 6 + c = 1, suy ra c = 1 (do a - -b). Do dd « = 4c - 3 = 1 va ^ = - 1 . 2
Vay ham sd can tim iay = x - x + 1 .
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• Do he sd a ^ 1 > 0 va gia tri nhd nha't ciia ham sd dat dupe tai x = — nen ham sd nghich bie'n tren khoang (-00 ; —) va dong bien tren khoang (— ; +co). Bang bien thien :
y=x
—00
+00
+00
+00
-x+1
Ham sd cd dd thi nhu hinh 2.11. b) Dudng thing y = mx cit parabol (?) tai hai didm M^A ; y/i) va B(xg ; yg) ndu va ehi ne'u phuong trinh X - X + 1 = mx hay x ^ - ( l +m)x+ 1 - 0 ed
hai
nghiem
phan
biet,
A = (1 + m)^-4 = m'-\-2m-3
(1) tiic
la
biet
thiic
duong.
Khi dd, hai nghiem ciia (I) chinh la x^ va x^. Theo dinh li Vi-et, ta ed
Hinh 2.11
x^ + Xg = 1 + m.
(2)
Tir (2) ta suy ra hoanh dp trung diem C eiia doan thing AB la _ x^ + Xg _ 1 + m 2 2
Xr- =
Do C la mot diem thude dudng thing id) nen tung dp y^ cua nd thoa man m(l + m) y^=.mxc = Ke't ludn. Toa dp trung diem ciia doan thing AB la C vdi dieu kien m + 2m - 3 > 0.
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52
I+m
m{\ + m)
2.37. Hien nhien ham sd y - 0 xac dinh vdi mpi x va cd dd thi ddi xiing qua true hoanh. Gia su ham sd y - / ( x ) xac dinh tren R, cd dd thi (G) nhan true hoanh lam true ddi xiing. Khi dd Vx G M : M(x ; y) e (G) <^ M'(x ; -y) e (G). Didu nay ed nghia la VxGE:y=/(x)^-^=/(x). Suy ra y = 0 vdi mpi x. vay ham sd y = 0 la ham sd duy nha't cd dd thi ddi xiing qua true hoanh. Chu y. Ciing cd the ehiing minh ring (G) triing vdi true hoanh. That vay, ne'u trai lai thi phai ed mot diem M(xo ; yo) thude (G) va yo T^ 0. Khi dd, do tinh ddi xiing qua true hoanh, didm M'(xo ; -yo) cung thude (G). Ta ed dudng thing MM' song song vdi true tung, cit (G) tai hai didm phan biet M va M' Dd la dieu khdng the xay ra ddi vdi dd thi cua mdt ham so. 2.38. a) F(-x) = ^ |/(-x) +/(x)] - ?(x). b) Gi-x) = i {fi-x) -fix)]
= - i Ifix) -fi-x)]
= -Gix).
2.39. a) A ( - l ; - m + 3), 5(3 ; 3m - 1). b) A nim o phia tren true hoanh khi va chi khi -m + 3 > 0, tu'c la m < 3. e) B nim cf phia tren true hoanh khi va chi khi 3m - I > 0, tu'c la m > - . d) Ca hai didm A va 6 ddu nim d phia tren true hoanh khi va ehi khi cae didu kien ndi trong cau b) va c) dong thdi dupe thoa man, nghia la - < m < 3. Khi dd, loan bd doan thing AB nim d phia tren true hoanh. Ndi each khae : (m - l)x + 2 > 0, Vx G [-1 ; 3] c^ - < m < 3. 2.40. a) y = -3(x - 1)^ - 3 ;
b) y - -3(x + 2)^ + 2.
2.41. Dudng thing y = -2,5 song song vdi true hoanh. Do dudng thing nay CO mot diem ehung duy nhat vdi parabol (?) nen diem ehung iy chinh la dinh ciia parabol (?). Tir do suy ra dinh I cua parabol (?) ed Uing dp y = -2,5. booktoan.com 53
Dudng thing y = 2 cung song song vdi true hoanh. Do dd trung didm C ciia doan thing AB nim tren true ddi xiing ciia parabol. Hoanh dd cua diem C la x =
-1 + 5
= 2. Vay true ddi xiing cua parabol la
dudng thing x = 2, suy ra hoanh dp dinh I cua (?) la x = 2. Toa dp cua / la (2 ; -2,5). Til dd suy ra ne'u (?) la dd thi cua ham sd 2
fix) = ax + bx + c thi / ( - I ) = a-b
+ c = 2, -—- = 2 va 2(3
A Ja
b^ - 4ac = -2,5. 4^
Tir dd suy ra a =—. b = -2 , c = -— va ham 1 2 1 • sd can tim la y = — x - 2x -— Dd thi cua ^ hinh 2 2 • ham sd nhu tren 2.12.
Hinh 2.12
2.42. Phuong an (B). 2.43. Phuong an (C). Chii y ring cac ham sd con lai ddu cd tap xac dinh khae R. 2.44. Phuong an (D). 2.45. Phuong an (A). Chi can chii y ring cin phai tinh tie'n sang trai. 2.46. Phuong an (D). Chu y. Tranh nhim lin ve phuong va hudng tinh tie'n. 2.47. Phuong an (C). Chii y. Tranh eac nhim lin vd da'u va nhim lin giua —2a . b va — a 2.48. Phuong an (D). 2.49. Phuong an (B). 2,50.(a)<^(3);(b)^(l). 2.51. (a) ^ (2) ; (b) ^ (3). 2.52. ( a ) < ^ ( 4 ) ; ( b ) < ^ ( l ) .
booktoan.com 54
phuang III •*
PHUONG TRINH
BAC NHAT
«
VA BAC HAI
A. NHONG KIEN THLJC CAN NHO Cac kie'n thUe dupe neu sau day cd bd sung mot vai ke't qua di nhan thay va dupe sic dung nhieu trong thuc hanh giai toan. 1. Cac phep bien doi tirong duong cua phuang trinh 1) Thuc hien eac phep bien ddi ddng nhat trong tCmg ve' nhung khdng lam thay ddi tap xac dinh eua phuong trinh. 2) Them vao hai ve' cua phuong trinh ciing mdt bidu thiic xac dinh vdi mpi gia tri ciia an thude tap xac dinh cua phuong trinh (trudng hpp hay dung la quy tac chuyen ve). 3) Nhan hai vd eiia phuong trinh vdi ciing mot bieu thiie xae dinh va khae 0 vdi mpi gia tri cua in thupc tap xac dinh cua phuong trinh (chii y ring chia cho mot sd tiCc Id nhan vdi nghich dao eua sd dd). 4) Binh phuong hai ve' eua mdt phuong trinh cd hai ve' ludn ciing dau khi in lay mpi gia tri thupc tap xac dinh cua phuong trinh. 2. Phep bien doi cho phuong trinh he qua Binh phuong hai vd cua mdt phuong trinh. 3. Giai va bien iuan phuong trinh dang ax +fa= 0 • £ 1 ^ 0 : phuong trinh ed mpt nghiem duy nha't x = • a^Ovab^O
: phuong trinh v6 nghiem.
• (3 = /> = 0 : phuong trinh nghiem diing vdi mpi x.
booktoan.com 55
4. Giai va bien luan phuong trinh bac hai mot an ax^ + bx + c = 0
(1)
2
2
vdi biet thde A = b ~ 4ac hay biet thiic thu gpn A' = 6' - ac (vdi b = 2b'). A < 0 (A' < 0) : (1) v6 nghiem. ' • A - 0 (A' - 0) : (1) cd mdt nghiem kep x = - —- X = 2a V A > 0 (A' > 0 ) : (1) cd hai nghiem phan biet x =
b' a
-b±yfK( 2(2
X =
I.
-/?' ± VA^
V
a
5. Djnh li Vi-et (thuan va dao) : 2
Hai sd Xi va xj la hai nghiem eiia phuong trinh bac hai ax + 6x + c = 0 khi vd chi khi ehiing thoa man hai he thiic Vi-et sau : _ b _c Xi + XT —
;
XIXT — — •
a Dinh li Vi-et cd the dupe ling dung dd : - Nhim nghiem ciia phuong trinh bac hai.
a
- Tim hai sd biet tdng va tich cua ehiing : Ne'u hai sd cd tdng bing S va tich bing P thi hai sd dd la hai nghiem ciia phuong trinh x^ -Sx-¥P
= 0.
(Tat nhien, didu kien tdn tai eua hai sd ndi tren la S^ - 4P > 0.) - Phan tich mpt tam thiic bac hai thanh nhan tii : Cho tam thiic bac hai fix) = ax + 6x + c. Neu phuong trinh bac hai fix) = 0 cd hai nghiem (cd thd triing nhau) xi va X2 thi tam thiic bac hai fix) cd thd phan tich dupe thanh nhan tii nhu sau: 2
ax + bx +c = aix-x{)ix-X2). - Tmh gia tri cae bidu thde ddi xiing ciia hai nghiem ciia phuong trinh bac hai : o = X] + X-1 = — — ', r = X1X9 — — j
' x\+ x\^^ 56
'^ a ^^ -2P; x\+ x\=S^
booktoan.com
a - 3PS.
- Xet da'u cae nghiem cua phuong trinh bac h a i : Phuong trinh cd hai nghiem trai dau o P <0. Phuong trinh ed hai nghiem duong <^ A > 0, /* > 0 va 5 > 0. Phuong trinh cd hai nghiem am o A > 0, /^ > 0 va 5 < 0. 6. Giai va bien luan he hai phuong trinh bac nhat hai an iax + by = c \a' X + by = C D-
D.-
^y
=
a
b
a'
b'
c
b
c
b'
a
c
a'
c'
i o ? ? (a + />^ ?^ 0 va d^ + ^'^ ^ 0).
(2)
= ab' - db ;
= cb' - c'b ; = ac - ac.
• D ?^ 0 : (2) cd mpt nghiem duy nha't (x ; y), trong dd x=—^,
y =-j^
• D = 0, D^ ?t 0 hoac D^ ?t 0 : (2) v6 nghiem. • D = D^ = D^ = 0 : (2) ed vd sd nghiem (x ; y) tinh theo edng thiic -by + c X e a (ne'u a^O) hoac • -ax + c y =
X = —
ineubitO).
Chiiy Khi giai va bien luan he phuong trinh ed chiia tham sd dang \ax + by = c lO'x + b'y
= c'.
cd thd xay ra trudng hpp a = b = 0 (hoac d = b' = 0). Khi dd, ta sir dung , cae kdt luan dd thiy sau day : - Phuong trinh Ox + Oy = c vd nghiem ndu c ^ 0, nghiem diing vdi mpi x va vdi mpi y ndu c = 0.
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- Trong mdt he phuong trinh, neu mpt phuong trinh eua he v6 nghiem thi he v6 nghiem. - Trong mpt he hai phuong trinh, neu mpt phuong trinh eiia he nghiem diing vdi mpi gia tri cua cae in thi tap nghiem cua he phuong trinh dd triing vdi tap nghiem cua phuong trinh con lai. , 7. Giai he phuOng trinh bac hai hai an
1) He phuong trinh trong dd ed mpt phuong trinh bac nhat : Diing phuong phap the. 2) He phuong trinh ma mdi phuong trinh trong he khdng thay ddi khi thay the' dong thdi x bdi y va y bdi x : Diing phuong phap dat an phu S = X + y ; P = xy.
B. DE BAI §1. DAI C U C ^ G Vt P H U O N G TRINH 3.1. Tim didu kien eiia mdi phuong trinh sau rdi suy ra tap nghiem cua nd : a) X - V x - 3 = V3-X + 3 ;
b) V-x^ + 4 x - 4 - = x^ - 4 ;
e) Vx - Vl - X = V-x - 2 ;
d)x + 2Vx + l = : l - V-x - 1.
3.2. Tim nghiem nguyen eua mdi phuong trinh sau bing each xet didu kien xac dinh cua nd : a) sl4-x
-2= y[x -x\
b) 3 Vx + 2 - V2 - x +2V2
3.3. Giai cac phuong trinh sau : a) X + Vx - Vx - 1 ;
b) x^ + V2 - x = V2 - x + 9.
3.4. Trong cac phep bien ddi sau; phdp bien ddi nao cho ta phuong trinh tuong duong, phep bie'n ddi nao khdng cho ta phuong trinh tuong duong ? 7 a) Lupe bo sd hang r o ca hai ve cua phuong trinh .A.
i
2 , 7 ^ 7 X + 1 + x - 1 = 2x + X - 1 '
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58
b) Luoc bo sd hang
d ca hai ve' eua phuong trinh x-2 x^ + 1 + - ^ x-2
=2x+
^ X- 2 '
c) Thay the' ( V 2 x - l ) bdi 2x - 1 trong phuong trinh ( V 2 x - l ) =3x + 2 ; 2
,
d) Chia ca hai ve' qua phuong trinh x + 3 = x + 3 eho x ; x^ + 1 1 e) Nhan ca hai ve' eiia phuong trinh =2 + — vdi x. X
X
3.5. Trong cac phep bidn ddi neu trong bai tap 3.4, phep bie'n ddi nao eho ta phuong trinh he qua ? 3.6. Kidm tra lai ring cac bien ddi sau day lam mat nghiem eua phuong trinh : ,
,
2
a) Chia ca hai ve cua phuong trinh sau cho x - 3x + 2 (x + l)(x^ - 3x + 2) = x^ - 3x + 2 ; b) Chia ca hai ve' ciia phuong trinh sau eho Vx - 1 (x + 4 ) V x ~ l = ( V x - l ) . 3.7. Giai eac phuong trinh sau bang each binh phuong hai ve': a)|2x + 3| = l ; b) | 2 - x | = 2x - 1 ; c) V 3 x - 2 - 1 - 2 x ; d)V5~2x = Vx - 1. 3.8. Tim didu kien xac dinh ciia phuong trinh hai in sau rdi suy ra tap nghiem ciiand V - x 2 - ( y + l)^+xy = (X + l)(y + 1).
§2. P H U O N G TRINH
BAC N H A T
VA
BAC
HAI
M6T
XN
Phuang trinh bac nhat 3.9. Tim cac gia tri ciia p dd phuong trinh sau v6 nghiem ( 4 / - 2 ) x = l + 2p-x.
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3.10. Tim cae gia tri ciia q di mdi phuong trinh sau cd v6 sd nghidm : a) 2qx ~ \ =x + q ;
b) q^x - q^ 25x - 5.
3.11. Tim cae gia tri eiia m de mdi phuong trinh sau chi cd mdt nghiem : a) (x - m)(x - 1) = 0 ;
b) m(m - l)x = m - 1.
'3.12. Giai va bien luan cae phuong trinh sau theo tham sd m : a) 2mx = 2x + m + 4 ;
b) m(x + m) = x + 1.
Phuang trinh bac hai 3.13. Vdi mdi phuong tiinh sau, bie't mdt nghiem, hay tim tham sd m va nghiem con lai: a) (2m^ - 7m + 5)x^ + 3mx - (5m^ - 2m + 8) = 0 cd mdt nghiem la 2. b) (5m^ + 2m - 4)x^ - 2mx - (2m^ - m + 4) = 0 cd mdt nghiem la - I . 3.14. Giai va bien luan cac phuong trinh sau theo tham sd m : a) mx + 2x + 1 = 0 ; b) 2x^ - 6x + 3m - 5 = 0 ; c) (m + l)x^ - (2m + l)x + (m - 2) = 0 ; d) (m^ - 5m - 36)x^ - 2(m + 4)x + 1 = 0. 3.15. Tim cac gia tri cua tham sd m di mdi phuong trinh sau cd hai nghiem bing nhau : a) x^ - 2(m - l)x + 2m + 1 - 0 ; b) 3mx^ + (4 - 6m)x + 3 (m - 1) = 0 ; e) (m - 3)x^ - 2(3m - 4)x + 7m - 6 = 0 ; d) (m - 2)x - mx + 2m - 3 = 0. 3.16. Bien luan sd giao diem cua hai parabol sau theo tham sd m : 2
X
2
y = x + mx + 8 va y = x + x + m. Dinh li Vi-et 3.17. Vdi mdi phuong trinh sau, biet mpt nghiem, tim m va nghiem con lai : 2
a) X - mx + 21 = 0 cd mpt nghiem la 7 ; b) X - 9x + m = 0 cd mpt nghiem la - 3 ; c) (m - 3)x^ - 25x + 32 = booktoan.com 0 ed mpt nghiem la 4. 60
3.18. Gia SIX Xi, x^ la cac nghiem cua phuong trinh 2x - 1 l x + 13 - 0. Hay tinh : a ) X, + X2 ; \
4
b ) x\
4
+ X2 ;
^)t(^^'^^)^?^-^')-
c) Xi - xJ ;
3.19. Gia sii xj, X2 la cae nghiem ciia phuong trinh x + 2mx + 4 = 0. Hay tim ta't ca cac gia tri eua m dd cd ding thUc : ( ^ \'
+ ^•^2
= 3.
\^\ J
3.20. Tim tat ca cac gia tri ciia a de hieu hai nghiem cua phuong trinh sau bang 1 2 x ^ - ( a + l)x + a + 3 = 0. 3.21.
Gia su Xiva X2 la cac nghiem cua phuong trinh bac hai ax + 6x + c = 0. Hay bieu didn cac bidu thde sau day qua eac he sd a,bvac a) x\ + x\ ;
b ) Xj^ + x ^ ; d ) Xj - 4x|X2 + X2.
Xi
Xo
3.22. Tim tit ca cac gia tri duong cua k de cac nghiem cua phuong trinh 2x- -ik + l)x + l = k^ trai da'u nhau va ed gia tri tuyet ddi la nghich dao cua nhau. 3.23.
Hay tim tat ca cac gia tri ciia k dd phuong trinh bac hai {k + 2)x^ -2kx-k
=0
ed hai nghiem ma sip xep tren true sd, ehung ddi xiing nhau qua diem x = l. 3.24. Gia sii a, b \a hai sd thoa man a> b>0. Khdng giai phuong trinh abx^ - (fl + &)x + 1 = 0, hay tinh ti sd giua tdng hai nghiem va hieu giua nghiem Idn va nghiem nho eiia phuong trinh dd.
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3.25. Giai eac phuong trinh sau day : a) X* - 5x^ + 4 = 0 ;
b) x^ - 13x^ + 36 = 0 ;
e) x"^ - 8x^ - 9 = 0 ;
d) x^ - 24x^ - 25 = 0.
3.26. Cac he sd a, b va c cua phuong trinh triing phuong ax + bx + c = 0 phai thoa man didu kien gi dd phuong trinh do a) V6 nghiem ?
b) Cd mpt nghiem ?
c) Cd hai nghiem ?
d) Cd ba nghiem ?
e) Cd bdn nghifm ?
§3. MOT S 6 PHI/ONG TRINH QUY vfi
PHUONG TRINH BAC N H X T H O A C
BAC HAI
3.27. Giai va bien luan cac phuong trinh sau theo tham sd a : a)
= a\ b) = a-3 x-1 x-2 3.28. Giai cae phuong trinh :
;
a) Vx^ + X + 1 = 3 - X ; b) Vx^ + 6x + 9 = |2x - l| ; c) x(x + 1) + x(x + 2) = x(x + 4) 1-x
1 + xy
U-x
/
14 - X
3.29. Giai cac phuong trinh : a) — - j - + x+1
,,2x-l
'^^ 62
^ - 1; X-2•
3x-l
x-7
x-1 3x 5 X ~ 2 x - 2 " ~ 2 " ' booktoan.com
c)
-— - 2. ax+ 3
3.30. Giai cae phuong trinh : 4x 5x a) -z + x^ + X + 3 x^ - 5x + 3
3 2 '
x-1 x-2 x-4 x-5 x +2 x+3 x+5 x+6 3.31. Giai va bien luan cac phuong trinh sau theo tham sd m : b)
a) I 3mx - 1 I = 5 ;
b) | 3x + m | ^ I 2x - 2m | ;
3.32. Giai va bien luan cac phuong trinh sau : a) (x - 2)(x - mx + 3) =^ 0 ; , , (x + l)(mx + 2) b)
T X — 3m
= 0 ; 2
mx - 1 m _ m{x + 1) x - 1 "^ x + 1 " j.^ _i 3.33. Cho tam giac ABC nhpn cd canh BC = a, dudng cao AH = h. Mdt hinh chU nhat MNPQ npi tie'p trong tam giac iM & AB • N ^ AC ; P, Q G BC) cd ehu vi bing 2p {p la dp dai eho trudc). Hay tfnh dp dai canh PQ cua hinh ehu nhat MNPQ, bien luan theo p, a, h.
§4. P H U O N G TRINH VA HE
PHUONG
TRINH
BAG N H X T N H I £ U A N
3.34. Xet tap hpp cac didm ed toa d6 (x ; y) la nghiem ciia phuong trinh ax + by = c. Tim dieu kien ciia a, b, c di : a) Tap hpp didm dd la mpt dudng thing di qua gd'c toa dp ; b) Tap hpp didm dd la mpt dudng thing song song vdi true tung ; c) Tap hpp diem dd la mpt dudng thing song song vdi true hoanh ; d) Tap hpp didm dd la true tung ; e) Tap hpp didm dd la true hoanh ; g) Tap hpp dd la mpt dudng thing cit hai true Ox va Oy tai hai diem phan biet.
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63
3.35. Giai eac phuong trinh sau va minh hoa tap nghiem tren mat phing toa dp : a)2x + 3y = 5 ;
b) 0.x + 3y - 6 ;
c) 2x + O.y = 4 ;
d) 2x + 3y = 0.
3.36. Giai va bien luan eac phuong trinh sau theo tham sd m : a) mx + (m - I)y = 5 ;
b) mx + my = m + 1.
3.37. Bing dinh thde, hay giai cac he phuong trinh sau :
'3x + 2y = -7 5x - 3y = 1 ;
a)
b)
'V2x + 4y = 1 2x + 4Sy
= 5.
3.38. Tinh nghiem gin dung cua cac he phuong trinh sau (chinh xac de'n hang phin tram) : jVSx + V3y = V2
|3x + (Vs - 2)y = 1
[V2x - V3y = V5 ;
[(V^ - l)x + ^^y = V5.
3.39. Giai va bien luan cae he phuong trinh theo tham sd a : ax + 2y - 1 X + (« - l)y - a.
a)
b)
ia - l)x + (2a -3)y = a ia + \)x + 3y = 6;
c)
d)
ia- 2)x + ia --4)y = 2 ia + l)x + i3a + 2)y = - 1 ; 3(x + y) _ ^ x-y .., 2x--y-^
3.40. Giai eac he phuong trinh :
j3|x| + 5 y - 9 = 0
"^\2x-\y\ = l;
b)
ixl - (2 = 1 ia la tham sd). y-2x = 5
3.41. Giai he eac phuong trinh ^ + ^.3 a) •
64
X
9 .^
y
10 y ~ '
b)
booktoan.com
=3 X - 2y + X + 2y 3
4
X - 2y
X + 2y
= -1.
3.42. Mpt ca no chay tren sdng trong 8 gid, xu6i ddng 135 km va ngupe ddng 63 km. Mpt lin khae, ea nd cung chay tren sdng trong 8 gid, xu6i ddng 108 km va ngupe ddng 84 km. Tinh van tde ddng nude chay va van toe ciia ca no (bie't ring van toe that cua ca nd va van tde ddng nude chay trong ea hai lin la bing nhau va khong ddi). 3.43. Cho hai dudng thing (df,) : (m - l)x + y = 5 va (0^2) : 2x + my = 10. a) Tim m dd hai dudng thing (Jj) va (.^2) cit nhau. b) Tim m di hai dudng thing (Jj) va (fl'2) song song. e) Tim m de hai dudng thing (rfj) va (^2) trung nhau. 3.44. Cho ba dudng thing idy) :2x + 3y = -4 ; id2):3x +
y=l;
id^) : 2mx + 5y = m. a) Vdi gia tri nao cua m thi (d^), id2), id^) ddng quy tai mpt didm ? b) Vdi gia tri nao eiia m thi ((^2) va id^) vudng gde vdi nhau ? 3.45. Vie't phuong trinh cua dudng thing trong mdi trudng hpp sau :, a) Cat true Ox tai didm ed hoanh dp la 5 va cit true Oy tai didm ed tung dd la - 2. b) Di qua hai didm AH ; - 1 ) va S(3 ; 5). 3.46. Giai cac he phuong trinh bac nhat ba in : 2x + y + 3z - 2
X + y = 25 a)
y + z = 30
b)
-X + 4y - 6z = 5 5 x - v + 3z - - 5 .
2 + X - 29 ;
3.47. Su dung may tinh bo tiii dd tim nghiem gan diing cua he phuong trinh sau (chinh xac de'n hang phin tram) : '4x + V2y + z = 1 a)
V3x + V3y + 2z = V2 X + V5y + 3z = V3 ;
[(V2 + l)x + y + V3z = - 1 b)
V . + V2y + V5z - V2
V3x + (V3 + l)y - z - Vs.
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65
3.48. Cd ba Idp hpe sinh lOA, \0B, IOC gdm 128 em cung tham gia lao ddng trong cay. Mdi em Idp lOA trdng dupe 3 cay bach dan va 4 cay bang. Mdi em Idp 106 trdng dupe 2 cay bach dan va 5 cay bang. M6i em Idp IOC trdng dupe 6 cay bach dan. Ca ba Idp trdng dupe la 476 cay bach dan va 375 cay bang. Hoi mOi Idp cd bao nhieu hpc sinh ? 3.49. Bdi todn co. Hay giai bai toan dan gian sau : Em di chaphien Anh gvti mot tien Cam, thanh yen, quyt Khdng nhieu thi it Mua du mot tram Cam ba dong mdt Quyt mot dong nam Thanh yen tuai tdt Nam dong mot trdi
Hoi mdi thii mua bao nhieu trai, bie't ring mpt tien la 60 ddng ?
§5 M 6 T S 6 vi DU V£ H£ PHUONG TRINH BAC HAI HAI XN Giai cae he phuong trinh sau
3.50. a)
'2x - y - 7 = 0
'4x + 9y = 6
y^ - x^ + 2x + 2y + 4 - 0 ; c)
b)
3x^ + 6xy - X + 3y = 0 ;
2x^ + X + y + I = 0 x^ + 12x + 2y + 1 0 - 0 . (x + y + 2)(2x + 2y - 1) - 0 3x^ - 32y2 + 5 - 0 ;
'(x + 2y + l)(x + 2y + 2) = 0 b) xy + y^ + 3y + 1 = 0. r3(x+.y) = xy [x^ + y^ = 160 ;
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5-BTDSlO.NC - B
3.53.a)|'"'-^'=^ xy + x^ = 2 ; c)
, J x 2 + y^=25-2xy [y(x + y) = 10 ;
2(x + y ) 2 + 2 ( x - y ) 2 = 5ix^ - y^) x^ +y^ = 20.
BAI TAP 6 N TAP CHUONG III 3.54. Phuomg trinh dang ox + ^ = 0 (in x) v6 nghiem trong trudfng hop nao, ed vd sd nghiem trong trudng hpp nao ? Ap dung. Tim cac gia tri ciia tham sd m sao cho phuong trinh mim - 2)x = m a) Cd nghiem duy nha't;
b) Vd nghiem ;
e) Cd vo sd nghiem ;
d) Cd nghiem.
3.55. Cho he phuong trinh (ax + by = c (I) <
[fl'x + b'y = C
, (an la X va y) thoa man dieu kien db'c' ^ 0. '•
Chiing minh ring : a) Ne'u —1^ ?^ — thi he (I) cd nghiem duy nha't. a b ' b) Neu —r = T7 5^ - r thi he (I) v6 nghiem. a b c ' c) Neu — = -— = —- thi he (I) cd vd sd nghiem. a b c ' Ap dung. Tim eac gia tri cua tham sd a sao eho he phuong trinh Ua + l)x + 3y ^ a [x + ia- l)y = 2 CO vd sd nghiem.
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3.56. Giai va bien luan cae phuong trinh theo tham sd m : a) (2m - l)x - 2 = m - 4x ;
b ) m l x - l ) + 1 = - (4m + 3)x;
2
c) m(x + 1) = m - 6 - 2x. 3.57. Giai va bien luan cac phuong trinh theo tham sd m : , (2m - l)x + 2 a ^^ '-m+1 x-2
(m-l)(m + 2 ) x ^ ^ ^ ^ ^ 2x + l
3.58. Giai cae he phuong trinh X + y = 16
2x - y + 3z = 4 a) \3x - 2y + 2z = 3
b)
5x - 4y - 2 ;
c)
y + z = 28 z + X = 22 ;
|x - y| = V2 2x-y =-1.
3.59. Cho he phuong trinh
(m - l)x + (m + l)y = m (3 - m)x + 3y = 2.
a) Tim eac gia tri eiia m dd he phuong trinh ed nghiem. Khi dd, hay tmh theo m cae nghiem eua he. b) Tim nghiem gin dung cua he, chinh xac de'n hang phin nghin khi
m - VS - 2. 3.60. Giai va bifn luan cae phuong trinh theo tham sd m : a)t2x + mj = |2x + 2 m - l | ;
b)|mx+l [ = | 2 x - m - 3 | ;
e) (mx - 2)(2x + 4) = 0. 3.61, Giai cae phuong trinh a) 1 +
2
10
50
x-2
x+ 3
( 2 - x)(x + 3)
b)
x^ - X - 1 2 L_L =2x. X -3
3.62. Sir dung dd thi dd bien luan sd nghiem eiia cae phuong trinh sau theo tham s6 k : a) 3x^-2x 68
= k-
booktoan.com
b ) x ^ - 3 | x l - j t + 1 =0.
3.63. Cho ham sd y - x^ + x - 2 cd dd thj la parabol iP), ham s6 y = 3x + k cd dd thi la dudng thing id). a) Hay bien lua.n sd nghiem eiia phuong trinh x + x - 2 = 3x + /:, tCr dd suy ra sd didm ehung cua parabol iP) va dudng thing id). b) Vdi gia tri nao ciia k thi dudng thing id) cit parabol (P) tai hai didm nim d hai phia khae nhau eiia true tung ? e) Vdi gia tri nao cua k thi dudng thing id) cit parabol iP) tai hai diem phan biet Of vd ciing mpt phia eua true tung. Khi dd hai didm ay nim d phia nao cua true tung ? 3.64. Cho hai phuong trinh x^ - 5x + ^ = 0 (1) va x^-lx
+ 2k = 0 (2).
a) Vdi gia tri nao ciia k thi phuong trinh (1) ed hai nghiem va nghiem nay gip ddi nghiem kia ? b) Vdi gia tri nao eiia k thi phuong trinh (2) cd hai nghiem X] va X2 thoa man x^ + X2 = 25 ? e) Vdi gia tri nao eua k thi ea hai phuong trinh ciing cd nghiem va mpt trong cac nghiem eua phuong trinh (2) ga'p ddi mdt trong cae nghiem eiia phuong trinh (1) ? 3.65. Giai cac he phuong trinh sau : a)
b)
c)
d)
2x^ - xy + 3y^ = 7x + 12y - 1 ;c - y + 1 = 0 ; [(2x + 3y - 2)(x - 5y - 3) - 0
[x-3y = i; x^ + y^ + 2x(y - 3) + 2y(x - 3) + 9 = 0 2(x + y ) - x y + 6 = 0 ;
'x^-2y^ =lx
^ ,^.,
^,
y^ - 2x^ =ly.
3.66. Cho he phuong trinh
\x^ + y^ - 2(a + 1)
[ix-^yY=4. a) Giai he phuong trinh vdi a = 2. b) Tim cac gia tri cua a di he cd nghiem duy nha't.
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69
Gidl THifiU MOT S 6
GAU HOI TRAC NGHlfiM K H A C H
QUAN
Trong cdc bdi tu: 3.67 den 3.71, hay chon phuang.an tra ldi diing trong cdc phuang dn dd cho. 1
O
/'3
3.67. Didu kien xac dinh eua phuong trinh x + • . = V2x + 4
X
la
3 (B) x > - 2 , x ^t 0 va x < - ;
iA) X > -2va X ^ 0 ; 3 (C) X > -2 va X < — ;
(D) Khdng phai cae phuong an tren.
3.68. Cap (x; y) = (1 ; 2) la nghiem eua phuong trinh (A) 3x + 2y = 7 ; (C) 0.x + 3y = 4 ;
(B) x - 2y - 5 ; (D) 3x + O.y = 2. f 3x + 4y = -5 3.69. Nghiem ciia he phuong trinh la I-2x + y = -4 (\_ -7 (A)(l;-2); (B) 3' 4 J'
(C)f-i;-5|;
(D)(-2;l).
3.70. Cho phuong trinh bac hai ox^ +/7x + c = Oed hai nghiem x^,X2 cung khae 0. Phuong trinh bac hai nhan —va — lam nghiem la : X,
X2
(A) cx^ -rhx + a = 0;
(B) 6x^ + ax + c = 0;
(C) cx^ + £7x + /? = 0;
(D) ax^ + ex + Z> = 0.
- - , „ , „ . . , , ,
. , (m + l)x - 1
3.71. Tap nghiem cua phuong tnnh (A)5={^}; (C) 5 = R ; 70
, = 1 trong trudng hpp m ^^^ 0 la
(B)S=0;
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(D) KhOng phai cac phuong an tren.
Trong cdc bdi 3.72 va 3.73 , hay ghep mdi dong 6 cot trdi vdi mdt dong a cdt phai deduac mot khdng dinh diing. 3.72. Cho phuong trinh x^ + 2mx + m^ - 2m - 1 = 0. a) Ne'u m> —
1) thi phuong trinh da eho v6 nghiem. 2) thi phuong tnnh da cho cd v6 sd nghiem.
b) Ne'u m < —3) thi phuong trinh da eho ed mpt nghiem kep. c) Ne'u m = —
3.73. Cho he phuong trinh
4) thi phuong trinh da cho cd hai nghiem phan biet. mx + 9y = 6 X + my = -2. 1) thi he phuong trinh da cho v6 nghiem.
a) Ne'u m = 3 b) Ne'u m = - 3 c) Ne'u m^±3
2) thi he phuong trinh da cho cd mot nghiem. 3) thi he phuong trinh da cho cd v6 sd nghiem. 4) thi he phuong tiinh da eho nghiem dung vdi mpi gia tri ciia hai in.
C. DAP SO - HUONG DAN - LOI GIAI 3.1. a ) 5 - {3} ;
b ) ^ - {2}.
c) Khdng cd sd thue nao thoa man ddng thdi hai didu kien x > 0 va -X - 2 > 0. vay phuong trinh v6 nghiem. d) Phuong trinh vd nghiem. 3.2. a) Didu kien xac dinh ciia phuong trinh la 0 < x < 4. Thii true tiep cac gia tri eiia x thupc tap {0 ; 1 ; 2 ; 3 ; 4} vao phuong trinh, ta tha'y phuong trinh ed cae nghiem x - 0 ; x = 4 v a x - 2 . b) Dieu kien xac dinh cua phuong trinh la - 2 < x < 2. Thu true tiep cac gia tri eua x thupc tap {-2 ; - 1 ; 0 ; 1 ; 2 } vao phuong trinh, ta thay phuong trinh cd mot nghiem x = 0.
booktoan.com 71
3.3; a) Vd nghiem. 3.4. a) Khong ;
b)x = - 3 . b) Co ;
e) Khdng ;
d) Khdng ;
e) Khdng.
3.5. a), b), c) vae). 3.6. a) Ta tha'y khi x = 1 hoac x = 2 thi x^ - 3x + 2 = 0. Do dd x = 1 va x = 2 la hai trong cae nghiem ciia phuong trinh da eho. Nhung sau khi bie'n d6i, ta dupe phuong trinh x + 1 = 1 ; phuong trinh nay khdng nhan x = 1 va X = 2 lam nghiem. b) Sau khi bie'n ddi, ta dupe phuong trinh i x + 4) = (Vx - 1 )^ Phuong trinh nay khdng nhan x = 1 lam nghiem, trong khi x = 1 la nghiem eiia phuong trinh ban dau. Chu y. Hai bai loan tren eho tha'y : Neu chia ca hai ve' cua mpt phuong trinh eho mpt bieu thiic thi cd thd lam mat nghiem eiia phuong trinh. 3.7. a ) x = - l vax = - 2 ;
b)x=l;
c) V6 nghiem ; d ) x = 2.
3.8. Didu kien ciia phuong trinh la - x^ - (y + 1)^ > 0 hay x^ + (y + 1)^ < 0. Didu nay tuong duong vdi x = (y + 1) - 0, tiic la (x ; y) = (0 ; - 1 ) (vi ndu trai lai, em hay chiing minh rang ta ludn ed x + (y + 1) > 0). Thu true tie'p X ~ 0 va y = - 1 vao phuong trinh, ta tha'y cap sd (0 ; - 1 ) diing la nghiem ciia phuong trinh da eho). 3.9.
p = ^
3.10. a) 2^x - 1 = X + ^ o (2 x - m = 0 hoac x - l = 0 c^ x~m Vay phuong trinh ehi ed mpt nghiem khi m = 1. b) m ?^ 0 va m ?^ 1. 72
booktoan.com
hoac x = 1.
3.12. a ) T a c d : 2mx = 2x + m + 4 < ^ 2 ( m - l)x = m + 4
(1)
- Vdi m - I ^ 0 hay m ?^ 1, chia hai ve' cua (1) eho 2(m - 1) ta dupe m+4 X = 2(m - 1) - Vdi m -1 = 0 hay m = 1, phuong trinh (1) trd thanh 0.x = 5, vd nghiem. b) Phuong trinh cd nghiem duy nha't x = ~ im + 1) khi m ^ I, nghiem diing vdi mpi x khi m = \. 3.13. a) Do X = 2 la nghiem ndn thay vao phuong trinh ta dupe : 4(2m^ - 7m + 5) + 6m - (5m^ - 2m + 8) = 0 hay 3m^ - 20m + 1 2 - 0 . Giai phuong trinh tren (in la m) ta ed ke't qua m e J6 ; —[. Vdi m = 6 , phuong trinh da cho trd thanh 35x^ + 18x-176 = 0 OQ
va cd hai nghiem la x, =2 vax2 = ' ^ • 2 Vdi m = —, phuong trinh da cho tro thanh 11 2
n
80
^
40 va CO hai nghiem la Xj =2 va Xj - " T T b) Vdi m = 1, nghiem thii hai la - ; vdi m - - - , nghiem thii hai la — 3.14. a) Ne'u m = 0 thi phuong trinh cd nghiem x=-— Ne'u m^O thi phuong trinh cd A' = 1 - m. + Ne'u 1 - m < 0 tde la m >1 thi phuong trinh da cho vd nghiem. + Ndu 1 - m = 0 tde la m = 1 thi phuong trinh da cho cd mpt nghiem kep X--1.
booktoan.com
73
+ Ne'u 1 - m > 0 tiic la m < 1 thi phuong trinh da cho ed hai nghiem phan biet
•1-Vl-m Xi =
. va X2 =
- l + VT^ m
m m vay, vdi m e ( - 00 ; 0) u (0 ; 1) thi phuong trinh cd hai nghiem
-1 - V l - m Xi =
. va X2 =
m
-1 + VT^ m m
Vdi m = 0, phuong trinh cd nghiem x = -— Vdi m = 1, phuong trinh cd nghiem kep x = - 1. Vdi m e (1 ; +QO), phuong trinh v6 nghiem b) Phuong trinh cd A' = 9 - 2(3m - 5) = - 6m + 19. Vdi m e
"2" ' + °° ' phuong trinh v6 nghipm.
19 3 Vdi m= —r. phuong trinh cd nghiem kep x = — 6 2 Vdi m e
(
19'l ~ ^ ' "2" ' phuong trinh cd hai nghiem
. 3 + Vl9-6m va X = 2 2 e) Vdi m = - 1 , phuong trinh cd nghiem x = 3. X=
3-Vl9-6m
Vdi m ^ - 1 , phuong trinh ed A = (2m + 1)^ - 4(m + l)(w - 2) = 8m + 9. Do dd, vdi m G ~ ^ ' ~ o" ' pbuong trinh vd nghidm. 9 Vdi m = - • - , phuong trinh ed mpt nghiem kep x = 5. 8 Vdi m e
(
9 ^ -— ; 1 w (1 ; +oo), phuong trinh ed hai nghiem _ 2m + 1 - VSm + 9 . _ 2m + 1 + V8m + 9 """ 2(m + l) ^^''~ 2(m + l)
booktoan.com 74
2
d) m - 5m - 36 = 0 o m = - 4 hoac m = 9 Vdi m = - 4, phuong trinh trd thanh Ox = 1 nen vd nghiem. Vdi m = 9, phuong trinh trd thanh -26x + 1 =: 0 nen cd nghiem x = — • Vdi m g {-4;9}, tacd A' = (m + 4) - (m - 5m - 36) = 13m + 52 Tir dd suy ra : Vdi m G ( -00 ; - 4 ] , phuong trinh v6 nghiem . Vdi m e ( - 4 ; 9) w (9 ; +oo), phuong trinh cd hai nghiem _ m + 4 - yji3im + 4) 9 X— m - 5m - 36
. _ m + 4 +Vl3(m + 4) „ va X — m - 5m - 36
• " ^v
Vdi m = 9, phuong trinh cd nghiem x = ——
26 3.15. a ) m e { 0 ; 4 } ;
b) m - ^ ;
11
..
c)m e < | - 2 ; - h
d) m =
14 ± 2V7 ^
3.16. Hoanh dd giao didm hai parabol la nghiem eua phuong trinh 2
2
X +mx + 8 = x + x + m. Phuong trinh tren tuong duong vdi phuong trinh (1 - m)x = 8 - m. Tir do suy ra : Neu m = 1 thi hai dd thi khdng ed didm ehung. Neu m ^i thi hai dd thi cd mpt diem ehung. 3.17. a) Gpi nghiem thii hai la X2. Theo dinh U Vi-et, ta ed : [7 + X2 = m [7x2 = 2 1 Giai he tren ta dupe X2 = 3, m = 10. b ) x 2 - 12 ;m= 32
-36. 29
booktoan.com
75
11 13 3.18. Theo dinh li Vi-et ta cd Xj + X2 = — ; X|.X2 = — (dd tha!y hai nghiem ddu duong). Do dd : 13 11
11
a) Xi +X2=(Xi+X2) - 3 x , X 2 ( x i + X 2 ) = l y ]
b ) xt + X2' - [ix, + X 2 ) ' - 2x1X2 f - 2xfxl
" ^-y-y =
=
473
g •
~ -
C) Xj - X2 = (Xi - X2){Xi + X2)t(Xi + X2) - 2 X]X2].
Tacd : 2 (X, -
2
I
X2) - ( X i + X 2 )
-
4X|.X2=> k
I -
Vl7 X2
=
- r -
Gia sirxj
Vi7 ^
X2
.,
4
4
759 / -
r — . D o d o Xi - X2 = — T ^ ^
'
1 VJ
AJ
Ddi vdi trudng hpp X] > X2, ta ed x^ - x^ = ~T2~"^ d)-
269 26
Gmy. ^ ( I - X 2 ] + ^ ( I - X ? ) = ^ ^I
X2 V
X-^\
'/
+ ^-2XIX2=^L1^_2XIX2.
X2
Xi
X,X2
'
3.19. m = ± V2 + Vs Gai y. Didu kien dd phuong trinh ed nghiem la : A' = m^ - 4 > 0 <:> Iwl > 2. fxi + X2 =
-2m
T h e o d i n h li V i - e t , ta c d X1X2 = 4 -i2
nen
r^ \ + v^2y
/
V-
^•^Ij
^
4 4 (Xj + X2) - 2 x ^ X 2 _ L X,1 +^^2X 2 22 2 X, X 1-^2 X1X2
(4m^-8)^ -2. 16
booktoan.com 76
-2.
Tacd
^x V ^-^2;
+ ^
= 3 o ( 4 m ^ ~ 8 ) - = 80 « ( m ^ - 2 ) ^ = 5 < » m ^ - 2 + V 5
=> m = ± V2 + V5 Cac gia tri nay ddu thoa man didu kidn \m\ > 2. 3.20. a e {-3; 9} Gai y. Didu kien dd phuong trinh ed nghiem la A = (<3 + 1)^ - 8(fl + 3) > 0 <^ fl^ - 6^ - 23 > 0.
(*)
Gpi hai nghiem eua phuong trinh da cho la xj, X2 (gia sir X2 > X|) a +\ Theo dinh li Vi-et ta ed
X, + x-^ = 2 Vl
-
2
a +3 2 '
2 2 Do X2 - X[ = 1 nen (x2 - X|) = (xj + X2) - 4x,X2 ^ 1, suy ra ^^^~-
-2ia + 3)=loa'^-
6a-
27 = 0 <^ a = 9 hoac a =-3
Rd rang ca hai gia tri nay deu thoa man (*) vi a - 6a - 23 ~ 4 > 0. 3.21. a) Xi +X2 =(xi+X2) - 2 x i . X 2 = — - 2 — = a ^ b)Xi
+X2
=(Xj+X2)
a
- 3 XiX2(Xi + X2) ^
a' . 1 1 X, + x-, c) — + — = -^ Xj X2 XiX ^ H2
d)xi^-4xiX2 + X2^-(xi+X2)^- 6 r i r 2 - ^ ^
6c _b^
-6ac
3.22. k = 3. Gai y. Gpi Xi, X2 la hai nghiem ciia phuong trinh . Ap dung dinh li Vi-et va theo yeu c^u bai toan ta cd X2 =
va
1 _ -t + 2 Xi + Xn — X| —
Xi Xo — Xi
—
r-p
7-k'
y-^i I booktoan.com 11
I'k^ TiJf — - — = -\tac6k
7 =9,dok>Qntnk
= 3.
w " ,k = 13 nghiem w ^ phuong u t^ 4 T , X2 = 5 + V4I cua tnnhurla xj = 5 - V — 7— Voi 3.23. Gpi X|, X2 la hai nghiem ciia phuong trinh : ik + 2)x^ -2kx - k = 0 thoa man yeu c^u bai loan. Khi dd —^-—- = 1 nen Xj + X2 = 2
Ngoai ra
2k 2k Xy + X-, = - — - nen - — - =2, dodo k = k +2. ^ ^ k +2 k +2 Suy ra khong tdn tai k thoa man bai toan. 3.24. Gpi xj, X2 la hai nghiem ciia phuong trinh sao cho x^ > X2 Khi do, do a > b > 0 nen Xi - X2 = V(Xi + X2)^ - 4x^X2 = / _ .
a + bi . ^
Xt + Xn =
4 ab
\(a - b\ VI ab I
a~b ab
a +b ab
Suy ra ti sd giua t6ng va hieu hai nghiem bang
a+b
a—b 3.25. a ) x = - ± l , x = ± 2 ; b ) x - ± 2 , x = ± 3 ; c ) x = ± 3 ; d ) x = ± 5 . 2
3.26. Dat y = x , ta ed phuong trinh bac hai ay^ + by + c = 0.
(I)
a) Phuong trinh triing phuong da eho vd nghiem ne'u va chi ne'u • Phuong trinh (1) vd nghiem, tiic la A = 6 - 4ac < 0, hoac • Phuong trinh (1) chi cd nghiem am, tiic la A > 0, a c > 0 va a& > 0 ->Ova--<0 a a J b) Phuong trinh triing phuong da cho cd mpt nghiem ne'u va chi ne'u phuong trinh (1) cd mdt nghiem y = 0, nghiem kia khdng duong, tiic la booktoan.com c -Ovaa&>0. 78
c) Phuong trinh triing phuong da eho ed hai nghiem neu va ehi ndu • Phuong trinh (1) cd mpt nghiem kep duong, tde la A = 0 va o/? < 0, hoac " Phuong trinh (1) cd hai nghipm trai da'u, tiic la ac < 0. d) Phuong trinh triing phuong da cho ed ba nghiem ne'u va ehi ne'u phuong trinh (1) cd mdt nghiem v = 0 va nghiem kia duong, tiie la c = 0 va ab < 0. e) Phuong trinh trung phuong da cho ed bdn nghiem n^u va ehi ne'u phuong trinh (1) ed hai nghiem duong, tiic la A > 0, oc > 0 va a6 < 0. 3.27. a) Didu kien : x^ \, dua phuong trinh ve dang ax = 3 + a
(1)
- Neu a = 0 thi (1) vd nghiem nen phuong trinh da eho v6 nghiem. -Neufl^Othi(l)c^ x Nhan tha'y a
3 + fl
a
=^ 1. Vay x = —— la nghiem eua phuong trinh da eho. ' a
b) Didu kien : x^2,
dua phuong trinh vd dang ia-3)x
= 4a-l.
(2)
- Ne'u a = 3 thi (2) cd dang Ox = 5 nen phuong trinh vd nghiem 4a-1 - Ne'u « ?^ 3 thi (1) <=> X = — Xet dieu kien x T^ 2, ta cd a-3 A
^
1
— ^2 <=> 4a-1^2a-6<^a^ a-3 Do dd, ndu a = — thi x = —^~^^\ l^?^-
— 2
Ket ludn. Vdi a = 3 hoac a = —, phuong trinh v6 nghiem. 1
Vdi a^3vaa^—,
.
V ,
,
. .
46f - 7
phuong tnnh co nghiem x ^ ———
c) Vdi (3^0, phuong trinh v6 nghiem. Vdi a^O, phuong trinh cd nghiem x =
3.28. a)x = l y
2a
booktoan.com 79
b) V u + 3)^ = | 2 x - l | o |x + 3 | = | 2 x - l l 2 <::> X + 3 = 2x - 1 hoac x + 3 = l - 2 x <:=>x = 4 hoac x = — . e) Bie'n d6i phuong trinh vd dang x(x - 1) = 0, do dd x = 0 hoac x = 1. d) Didu kien : x ^ ±1, x i^ 14, x ?^ 0. Ta ed : (1 + x)^ - (1 - x)^ 1 - X 1 _ ;^-2 • 2x
3 14 - X
2 1+X
3 14 - X
o 5x = 25 o X = 5 (thoa man didu kien). 3.29. a ) x = 2 ± Vd ;
b)x e j s ; - - ^ i ;
c)x e | 2 ; ^
3.30. a) Nhan tha'y x = 0 khdng phai la nghiem, nen phuong trinh da cho tuong duong vdi phuong trinh : 4
5
3
1
+
x+— +1
3
x+
X
3
5
^
2
X
3 Dat y = X + — ta nhan dupe phuong trinh
' ^A^-i y- 5 2
n
y+1
Bie'n ddi phuong trinh (*) thanh f^—-7^ — = 0. Phuong trinh nay ed (y + i)(y - 5) hai nghiem y^ = - 5 , y2 = 3. Tir dd din de'n hai trudng hpp sau : _ 3 •x+-=-5 ^ ^
fx^ + 5x + 3 = 0 [x ;t 0
3 _ [x^ - 3 x + 3 = 0 • X + — == 3 <=> -^ ^
80
- 5 ± Vl3 -^—. 2
^ phuong tnnh v6 nghiem.
X ?t 0
Ket luan. Phuong trinh cd nghiem x
b)xe
<^ x =
{^\-\\-
-5 ± vl3
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-4 2 3.31. a) Vdi m^O, phuong trinh cd hai nghiem x = -— va x = —. 3m m Vdi m = 0, phuong trinh vd nghiem. b) Vdi m = 0, tap nghiem 5 = {0}. vdi m^O, tap nghiem S = |-3m v-rf3.32. a) Vdi m = 1 hoac m = - , tap nghiem S= [2]. 5 f 3 Vdi m ?£ 1 va m ^ - , tap nghiem S = I2 ; m - 1 . b) Didu kien la x T^ 3m. Khi dd ta ed (x + l)(mx + 2) = 0 < = > x + l = 0 hoac mx + 2 = 0. i ) x + l = 0 < : : > x = - l . Xet didu kien x ^ 3m, ta ed : 3m = - 1 <=> m = - -3. Do dd ne'u m = - -3 thi x = - 1,' bi. loai. . ii) mx + 2 = 0 <=> mx = - 2.
(1)
- Ne'u m = 0 thi OJ: = - 2 (v6 li), phuong trinh nay vd nghiem. 2 - Ne'u m ^ 0 thi (1) cd nghiemx . m 2 2 ^ 3m vdi moi m ndn x = la nghiem eua phuong trinh Nhan tha'y m • m 2 = - 1 <=> m = 2. Tut dd ta cd ke't luan sau : da cho. Lai cd m Ket ludn 1
("2
1
Vdi m ?t 0, m 9t 2 va m ^ - - , tap nghiem la 5 = i ;- 1• [ m } 3 Vdi m = 0 hoac m = 2 , tap nghiem la 5 = { - 1 } . Vdi m - - - , tap nghiem la 5 = {6}. c) Vdi m 9^ — va m ^ 0, tap nghiem 5 = j^r
r( ; Vdi m = - hoac
m = 0, tap nghiem S = 0. Gai y. Didu kien la x ?^ ±1. Dua phuong trinh vd dang (2m - l)x = 1 + 2m.
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6-BTDS10,NC-A
81
3.33.DatPQ=^MN
= xiO < x < a)
Theo dinh li Ta-let ta cd (h. 3.1) MN BC
AI ( AHy
AN AC
h-IH
IH =
ia - x)h
Didu kien MA' + IH = p cho ta ia - x)h = p hay phuong tnnh x H
ia-h)x
= aip-h).
Hinh 3.1
(1)
- Ne'u a = h thi phuong trinh (1) vd nghiem khi p ^ h, nghiem dung vdi mpi X khi p = h. Dieu nay cd nghia la : + Khi tam giac nhpn ABC co AH =^ BC va p ^ AH thi khdng cd hinh chii nhat nao thoa man didu kien eiia bai toan. + Khi tam giac nhpn ABC cd AH = BC va p = AH thi cd vd sd hinh chfi nhat thoa man dieu kien bai loan vdi canh x (0 < x < a tuy y). - Neu a^ h thi phuong trinh (1) cd nghiem duy nha't x = —^ a- h Xet didu kien 0
(2)
a- h Vl a^h
nen cd hai trudng hpp :
+ Neu a > h,ta CO i2) <:> 0 < p - h < a - h <^ h < p < a. + Ne'u a < h,ta CO i2) <:> {) > p - h > a - h <;:> a < p < h. Didu nay ed nghia la, gia tri x = —^—~ a- h chi khi p nam giUa a va h. 3.34. a) a^ +b^ ^0, c)a =
0,c^O,b^O;
e)a = c = 0, 82
c = 0;
b^O;
la nghiem cua bai toan khi va
b)b =
0,a^O;c^O;
d)a^O,
b = c = 0;
booktoan.comg)a^O,
b^O,c^O. 6-BTDSlO.NC-B
3.35. a) y=
c)
5-2x
x =2
b)
X G
y = ^>
d)
veR
-2 y=
^x.
Minh hoa tap nghiem bang hinh 3.2 a, b, c, d.
yk
o b) yk 0 1 0
d)
c) Hinh 3.2 3.36. a) Ne'u m = 0 thi phuong trinh cd v6 sd nghiem N^u m = 1 thi phuong trinh cd v6 s6' nghiem
X G
y = -5.
'x = 5 VGR,
Ne'u m^Ovam^^
Ithi phuong trinh cd vd sd nghiem
5 - mx y= m- 1 X e
b) Phuong trinh v6 nghiem ne'u m = 0 ; cd v6 sd nghiem y= ne'u m=^0.
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m + 1 - mx m 83
3.37. a) Ta cd : 2 -1 3 D= = -19 ; D, = 5 -3 1
2 3 = 19 ; D^ = -3 5
-7 = 38. 1
D. . DV 38 Do d o . = ^ = - 1 ; , = ^ = — = - 2 . He phuong trinh cd nghiem duy nh^t (x ; y) = (- 1 ; - 2). b) He vd nghiem. fx w 0,24 x=l 3.38. a) ' (sir dung may tinh bd tui). b) \ y « 0,47 ; y « 1,23. 3.39. a) Ta ed D = (fl + \)ia - 2) ; D^ = - (a + 1); D^ = (A - \)ia + 1). -1 X = a-2 • Vdi a ?t - 1 va i7 5t 2 thi D ?t 0, he cd nghiem duy nha't a-\ " " a- 2' • Vdi fj = - 1, he da eho tuong duong vdi phuong trinh -x + 2y = 1 nen cd X e
vd sd nghiem y=
1+ x
f 2x + 2y = 1 • Vdi fl = 2, he tro thanh { nen v6 nghiem. x+y=2 X =
b) Vdi a ?t 0 va a ^ - , he ed nghiem duy nh^t
1
2a-\ -3 y= 2a-V
X €
Vdi a = 0, he cd v6 sd nghiem 3^ =
-1-x
Vdifl i^ - , he vd nghiem. X =
c) Vdi (7 ?t 0 ; a ?^ 2, he ed nghiem duy nha't
booktoan.com 84
2a a-3 y = 2a '
Vdi a = 0, he vd nghiem. Vdi <3 = 2, he vd sd nghiem { ' b = 2 - X. d) Didu kien : x^y.
Bie'n doi he phuong trinh vd dang : | ( 3 - a)x + (3 + fl)y = 0
l3x - 2y = a. Ta cd D = - a - 15 ; D^ = -«(3 + a);Dy = a(3 - a) a(3 + a) a+ 15 • Vdi a ?t - 15 thi D ?^ 0, he (I) cd nghiem duy nha't aja - 3) ^ y = ^ + 15 ' X -
xTi.«
1='
i
a(3 + a)
«((3 - 3)
Nhan thay rang -^——^ = —^——^ <=> A = 0, '' (7 + 15 a + 15 nen khi a ?^ 0 thi x # y, khi dd nghiem cua (I) cung la nghiem cua he da eho. • Vdi a = - 15 thi D = 0, D^ ?^ 0, Dy ?^ 0, he (I) v6 nghiem nen he da cho vd nghiem. Ke't luan. Vdi a^Ovaa^(x;y)-
15, he cd nghiem duy nha^t: ai3 + a) aia - 3) a + 15 ' a+ 15
Vdi a = 0 hoac a = - 15, he vd nghiem. 3.40. a)
'3|x| + 5 y - 9 = 0 ( l ) 2x - \y\ = 1.
(2)
Tiir (2) suy ra 2x = 7 + |y|, nen phai ed x > 0. 44
|'3x + 5y - 9 Neu y > 0, he ed dang { Khi dd 12x - y - 7
booktoan.com
X =
y =
13
(loai)
13 85
26
r3x + 5y = 9 Ne'u y < 0, he ed dang \ Khi dd [2x + y = 7
He ed nghiem duy nh^t (x ; y) =
-3
(thoa man)
26 ~3 W ' 7
b) |x| = a + 1 Ne'u a > ~i thi X = ± (<3 + 1), he cd hai nghiem \a ia + \ ; 2a + 1) vk
i-a-\
;3.-2a).
Neu a = -\ thi |x| ^ 0 <=> x = 0, he cd nghiem la (x ; y) = (0 ; 5). Neu « < -1 thi |x| = ii + 1 < 0, he v6 nghiem. 1 1 , fdw + 5v = 3 3.41. Dat ~ - u ; — = V, he da cho tro thanh { • X y • [9«-10v = l he nay cd nghiem duy nha't (« ; v) =
i i 3'5r
Tir dd nghiem ciia he phuong trinh da cho : (x ; y) = (3 ; 5).
b)(x \y) = i^; ^ l
Gaiy. Dat - \ - = u ; ~ ^ = v.
V 70 140 J X ~2y x + 2y 3.42. Gpi van tde ddng nude la x (km/h), van tde eiia cand la y (km/h). (Didu kieny >x > 0). Khi dd, van tde ca nO di xudi ddng la (y + x), van tde ea nd di ngupe ddng la (y - x). 135 Ta cd he phuong trinh
63
X+ y + y- X =8
108
84 + X+ y y- X
Giai he tim dupe x = 3 ; y = 24. Trd lai. Van tde that ciia ca nd la 24 km/h ; Van tde ddng nude la 3 km/h.
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86
3.43. Xet he phuong trinh
\m - l)x + y = 5 [2x + my = 10.
Ta cd D = (m + l)(m - 2), D^ = 5(m - 2) ; D,, = 10(m - 2). a) (J,) va ((^2) cat nhau <^D^0
<^ m^-
1 vam?^2.
b) idi) II id2) <^ D = 0 va D_, ?^ 0 (hoac D^, ?^ 0) c^ m = - 1. c) (t/|) va (t/2) triiug nhau <=>D = D^ = D^ = 0 c ^ m = 2. 3.44. a) id{), id2) va (J3) ddng quy khi va ehi khi he phuong trinh ; •2x + 3y = - 4 (1) 3x + y = 1
(2)
2mx + 5y = m (3) cd nghiem duy nha't. Giai he phuong trinh gdm (1) va (2) tim dupe x = 1 ; y = - 2. Thay vao (3) tim dupe m = 10. b) (^2) -L (^3) ^
( - 3 ) - ^ := - I o m =
^
3.45. a) 2x - 5y = 10; b) y - 3x - 4. 3.46. a) (x ; y ; 2) = (12 ; 13 ; 17). Gai y. Cdng ve' vdi vd eiia ba phuong trinh trong he, dan de'n X + y + z = 42. Tir dd dd dang suy ra x = 12 ; y - 13 ; z = 17. b)(x;y;2)-(-l;2;-).Gmy. 2x + y + 3z - 2
-X + 4y " 6z - 5 <=> • -3x + 2y - 7 5x - y + 3z ^ - 5
^ _2 ' ~ 3
-X + 4y - 6z = 5 o
X - - 1
ly = 2.
8y = 16
3.47. Sii dung may tinh bo tui : X ^ -0,42 a)
y «2,91 ^-1,45;
x^-1,18 b) y « 1,62 z«0,14.
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3.48. Gpi sd hpe sinh cua Idp lOA, lOB, IOC Idn lupt la x, y, z. (Didu kien : x,y,z nguyen duong) Theo dd bai, ta lap dupe he phuong trinh x + y + z = 128 3x + 2y + 6z = 476 4x + 5y = 375.
(1) (2) (3)
Diing phuong phap khii d^n in so : nhan hai ve' cua (1) vdi - 6 rdi edng vao phuong trinh (2), ta dupe he phuong trinh : x + y + z = 128 • 3x + 4y = 292 4x + 5y = 375. Tii hai phuong trinh cud'i tim dupe x = 40 ; y = 43. Tir dd th^ vao phuong trinh ddu tim dupe z = 45, (thoa man didu kien bai toan). vay Idp lOA cd 40 em, Idp lOB cd 43 em, Idp IOC cd 45 em. 3.49. Gpi sd cam, quyt, thanh yen Idn lupt la x, y, z qua. (Didu kien : x,y,z nguyen duong nhd hon 100). Theo dd bai, ta lap dupe he phuong trinh ';c + y + z = 100
(1)
3x + ^ + 5z = 60.
(2)
Ttc (1) va (2) suy ra 7x + 12z = 100 <=> 7(x - 16) = -12(z + 1). Vl vay
r x - 1 6 = -12)t ikeZ)o [z + 1 = 7yt
fx = -12^ + 16 \ [z = 7A:-l.
Dd X, y nguyen duong thi k = I. Til do tim dupe x = 4 ; y = 9 0 ; z = : 6 (thoa man didu kien bai toan). vay cd 4 qua cam, 90 qua quyt va 6 qua thanh yen. 3.50. a) The' y = 2x - 7 vao phuong trinh thtr hai ddn ddn phuong trinh bac hai
ri3 5^ eua X. Til dd he ed nghiem la —-; - va (3 ; -1). V 3 3y 6-4x r 14^ b) Tuong tu, the'y = — booktoan.com He cd nghiem la ( -3 ; 2) va I - 2 ; — 88
c) Nhan phuong trinh thii nha't vdi 2 rdi trit vao phuong trinh thir hai ta dupe 3 x^ - lOx - 8 = 0. Tii do he cd nghiem (4 ; -37) va
2_n 3' 9
3.51. a) He da cho tuong duong vdi fx + y + 2 = 0 3x^ - 32y^ + 5 = 0
'2x + 2y - 1 = 0 hoac
3x^ - 32y2 +5 = 0.
Tit dd giai tuong tu nhu bai 3.50 ta dupe nghiem la ( - 3 ; 1),
i;-i]v/
41 29 '
17 29
29 ' 5 8
b ) ( - 3 + 2V2 ; I - V 2 ) , ( - 3 - 2 V 2 ; 1 + V 2 ) ,
/ V.
;
Gai y. He da eho tuong duong vdi fx + 2y + 1 = 0
fx + 2y + 2 = 0
[xy + x'' + 3y + 1 = 0
[^y + x^ + 3y + 1 = 0.
3.52. a) (I ; 2 ) v a ( 2 ; 1). b ) ( - 5 - V55 ; - 5 + V 5 5 ) v a ( - 5 + V55 ; -5 - V55). c) (6 ; 9) va (9 ; 6). Gai y. Ta tim dupe x + y = 15, xy = 54 hoac x + y = - 16, xy = 85. 3.53. a) ( 1 ; - l ) v a ( - l ; -1). Gtr/y. Ta cd xy + x^ = 2(2x^ - y^). Suy ra (x-y)(3x + 2y) = 0. b) ( - 3 ; -2) va (3 ; 2). Gai y. Tir phuong trinh thu: nha't suy ra x + y = 5 hoaex + y = - 5. e)(3V2 ;V2),(3V2 ; - V 2 ) , ( - 3 V 2 ; - yf2 ) va {-3 yfi ;V2). Gai y. Ttr phuong trinh thii nha^t nit ra x = 3y hoac x = ~3y. 3.54. Phuong trinh dang ax + h = 0 v6 nghiem khi a = 0 va /? ?^ 0, ed v6 sd nghiem khi a = b = 0.
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Ap dung. Ddi vdi phuong trinh m(m -2)x = m, ta ed : a) Phuong trinh cd nghiem duy nhat ndu mim - 2) ?!^ 0 ; b) Phuong trinh vd nghiem ne'u m = 2 ; c) Phucmg trinh cd vd sd nghiem ne'u m = 0 ; d) Phuong trinh ed nghiem ne'u m - 2 ^ 0 (tde la m ^ 2). iax + by = c .. 3.55. Xet he phuong trinh (I) < (an la x va y) voi dieu kien [a'x + b'y = c' db'c' ^0. a) Ne'u —7 ?^ 77 thi D = ab' -db^O a b b) Ne'u -^ = a vd nghiem. a _ c) Neu ~ =
nen he (I) ed nghiem duy nha't.
-^ ^ -^ thi D = ab'~ db - 0 va D„ = cb' -c'b^O b c /? _ c — = — thi D = 0 va D^ = c/?' - c'b = D = ac' -dc
nen he (I)
= 0 nen
he (I) ed vd sd nghiem. Chu y. Ket qua tren van dung khi a = b~0. Ua + l)x + 3y = a Ap dung. Ddi vdi he phuong trinh \ , ta ed : [x + ia- \)y = 2 - Ne'u (3 = 1 thi de tha'y he ed nghiem duy nha't. -- , , ,^, , , ,. ,,. a +1 3 - Neu (3 ?t 1 thi he eo vo s6 nghiem khi —-— = ^ • I a-\ dupe a = -2. m + 2
3.56. a) X =—
0
(7 _.,. . = —. Giai ra ta 2
")
Gai y. (2m- - l ) x - 2 - m - 4 x < = > (2m'^ + 3)x - m + 2.
2m- + 3 b ) m ^ ( x - 1)+ 1 = - ( 4 m + 3)x o (m + l)(m + 3)x = m ^ - 1 . • Ne'u m ?^ - 1 va m ?^ - 3 thi phuong trinh cd nghiem duy nha't m-1 x= -• m+3 • Ne'u m - - 1 thi phuong trinh nghiem dung vdi mpi x e R. • Neu m = - 3 thi phuongbooktoan.com trinh vd nghiem. 90
e) m(x + 1) = m^ - 6 - 2x <=> (m + 2) X = (m + 2)(m - 3). • Ne'u m = -2 thi phuong trinh nghiem diing vdi mpi x G R. • Ne'u m Ti - 2 thi phuong trinh cd nghiem duy nha't x - m - 3. 3.57. a) Vdi didu kien x^2,
phuong trinh da cho tuong duong vdi phuong trinh ( m - 2 ) x = - 2 ( m + 2).
(1)
Ne'u m = 2 thi (1) v6 nghiem nen phuong trinh da cho v6 nghiefn. Ne'u m ?^ 2 thi (1) CO nghiem duy nha't x =
-2(m + 2)
, De la nghiem ciia
phuong trinh da eho, gia tri nay phai thoa man didu kien x^2,
tde la :
-2(m + 2 ) ^ ^ m-2 Didu dd xay ra khi va chi khi m ^ 0. Vay, ta ed ke't luan : • Ne'u m = 2 hoac m = 0 thi phuong trinh da cho vo nghiem. • Neu m ;t 2 va m ^ 0 thi phuong trinh da eho cd nghiem x -
-2(m + 2) m- 2
b) Didu kien la x ^ - — • Ne'u m^-2,m^
1 va m ?^ 3 thi phuong trinh ed nghiem x = m- 3
• Ne'u m = - 2 thi phuong trinh nghiem dung vdi mpi x ^ - —. • Neu m = 1 hoac m = 3 thi phuong trinh vd nghiem. 3.58. a) Vd nghiem ; b) (x ; y ; z) = (5 ; 11 ; 17). c ) ( - 1 - V2 ; - l - 2 > ^ ) v a ( - 1 + V2 ; - 1 +2V2). Gai y. Do ix - y| = V2 <=> X - y ^ ± V2 nen tap nghiem ciia he phuong trinh da cho bang hpp cac tap nghiem ciia hai he phuong trinh
f ^ - ^ - ^ va [ 2 x - y = -1
\^~y--^ [2x"y--l.
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3.59. a) Ta ed : m-1 D3- m D.-
m 2
m +1 = im- 2)(m + 3); 3 m+ 1 = m- 2; 3
D.=
m-1 3-m
m = (m-2)(m+l). 2
Td dd suy ra he ed nghiem trong hai trudfng hpp sau : • D 7^ 0, tire la m ;t 2 va m ^ - 3 . Luc nay, nghiem duy nha't ciia he la (x;y)
^ 1 m +1 m+3'm+3
(1)
• D = D^ = Dj, = 0, tiic la m = 2. Liie nay he c6 vd sd nghiem (x ; y), trong dd X = 2 - 3y, va y e R (tuy y). b) Khi m = V5 - 2, he phuong trinh cd mdt nghiem duy nha't tinh theo (l).Vay 1 x = V5 + 1 « 0,309, y=
V?-i V5 + 1
^ 0,382.
3.60. a) Dd giai phuong trinh | 2x + m | = | 2x + 2m - 1 j, ta giai hai phuong trinh sau : 2x + m = 2x + 2m - 1 2x + m = -(2x + 2 m - 1).
(1) (2)
(l)<^Ox = m - 1. Phuong trinh nay vd nghiem ne'u m ^ I va nghiem diing vdi mpi x ndu m = 1. ^'^^ ^
A
1
1
~3m + 1
• (2) <^ 4x = -3m + 1 <=> X == 4 Ke't luan -3m + 1 - Neu m 7t 1 thi phuong trinh da cho cd mpt nghiem x = Ne'u m = 1 thi phuongbooktoan.com trinh da eho nghiem dung vdi mpi x. 92
Cha y. Cung cd thd giai phuong trinh nay bang each binh. phuong hai ve : I 2x + m I = I 2x + 2m - 1 I o (2x + m)^ = (2x + 2m - 1)^ o 4 ( l -m)x = im-
l ) ( 3 m - 1).
b) Viec giai phuong trinh | mx + 1 j = | 2x - m - 3 | quy vd giai hai phuong trinh (m - 2)x = - (m + 4) va (m + 2)x = m + 2. Ke't ludn - Ne'u m ;t + 2 thi phuong trinh ed hai nghiem x =
va x = 1. 2- m
- N^u m = -2 thi phuong trinh nghiem diing vdi mpi x. - Ne'u m = 2 thi phuong trinh cd mpt nghiem x = 1. Chu y. Cling ed thd giai phuong trinh nay bang each binh phuong hai ve': I mx + 1 I = I 2x - m - 3 I c^ (mx + 1)^ = (2x - m - 3)^ <=>
(mx + 1 + 2x - m - 3)(mx + l - 2 x + m + 3) = 0
o [(m + 2)x -{m + 2)][(m - 2)x + m + 4] = 0. c) Ke't luan - Neil m^Ovkm^
2 -Ithi phuong trinh ed hai nghiem x= — va x = - 2 . m
- Ne'u m = 0 hoac m = - 1 thi phuong trinh cd mpt nghiem x = - 2. 3.61. a) Vdi didu kien x ^ 2 vk x ^ - 3, phuong trinh da cho tuong duong vdi phuong trinh (x - 2)(x + 3) + 2(x + 3) = 10(x - 2) + 50.
(1)
(1) <^ x^ - 7x - 30 = 0 <^ X = 10 hoac x = - 3. Ddi chidu vdi didu kien, chi cd nghiem x= 10 la thich hpp. b) Vdi didu kien x ^ 3, phuong trinh da eho tuong duong vdi phuong trinh x ^ - | x | - 12 = 2 x ( x - 3 ) h a y x^ + | x | - 6 x + 1 2 = 0.
(2)
• Neu X > 0 thi (2)
o x ^ - 5 x + 12 = 0,
phuong trinh nay v6 nghiem.
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• Ne'u X < 0 thi (2)
c^ x^ - 7x + 12 = 0 c^ X = 3 hoac x = 4 (ca hai bi loai do x < 0).
vay phuong trinh da eho v6 nghiem. 2
,
3.62. a) Ve parabol y = 3x - 2x va xet dudng thang y = it (h. 3.3), ta co : • Ne'u k < - - thi phuong trinh vd nghiem. • Ne'u ^ = - - thi phuong trinh ed mdt nghiem (kep). 1 • Ne'u k > - - thi phuong trinh ed hai nghiem phan biet. Chii y. Ke't qua tren ciing cd thd dupe kidm nghiem lai bang phuong trinh bac hai 3x - 2x - / : " 0, vdi biet thiic thu gpn la A' = 1 + 3k.
Hinh 3.3
b) Ve dd thi ciia ham sd y = x - 3| x | + 1 va dudng thang y = k ih. 3.4), tacd : • Ne'u k < - -- thi phuong trinh vd nghiem. • Ne'u k = - ~ thi phuong trinh cd hai nghiem (ca hai ddu la nghiem kep). Neu - - < ^ < 1 thi 4 phuong trinh ed 4 nghiem.
Hinh 3.4
• Ne'u k = 1 thi phuong trinh ed 3 nghiem. • Neu k > 1 thi phuong trinh cd 2 nghiem. Chii y. Cd thd kidm nghiem lai k^t qua tren bang each giai va bien luan phuong trinh da cho theo tham sd k.
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3.63. a) Ta cd X + x - 2 = 3x + yfe tuong duong vdi phuong trinh x ^ - 2 x - ( 2 + ^) = 0.
(1)
Phuong trinh bac hai (1) cd biet thu'c thu gpn A' = k+ 3.Do d6 : • Ne'u /: < - 3 thi A' < 0, phuong trinh (1) vo nghiem nen dudng thing id) va parabol (P) khdng cd didm ehung nao. • Ne'u ^ = - 3 thi A' = 0, phuong trinh (1) cd mdt nghiem nen dudng thang id) va parabol iP) cd mpt diem ehung. • Ne'u ^ > - 3 thi A' > 0, phuong trinh (1) ed hai nghidm phan biet nen dudng thang id) va parabol (P) cd hai diem ehung phan biet. b) Dudng thing id) cat parabol iP) tai hai diem nim d hai phia khae nhau cua true tung khi va ehi khi phuong trinh (1) cd hai nghiem trai da'u, nghia la - ( 2 + jt)<0, h a y / : > - 2 . e) Dudng thing id) cit parabol (/*) tai hai didm nam d ciing mpt phia ciia true tung khi va chi khi phuong trinh (1) cd hai nghiem ciing da'u. Ne'u gpi hai nghiem a'y la x^ va X2 thi x^ + X2 = 2 > 0. Didu dd ehdng to ring khi hai nghiem ciing da'u thi ehiing ed da'u duong, nghia la hai giao didm nim d ben phai true tung. Mudn vay, ta phai ed : A' = it + 3 > 0 va - (2 + Jt) > 0, tiic la - 3 < /: < - 2. 3.64. a) Didu kien dd phuong trinh (1) cd nghiem la A, = 25 - 4it > 0. Vdi didu kien dd, gpi hai nghiem eiia (1) la X] va X2. Theo didu kien cua dd bai, tacd : X| + X2 =5 Xi X T ^ fC
X-y '^^ Z.Xi .
TCr do suy ra k=-^.
Khi dd, (1) ed hai nghiem laxj = - vax2 - ^ •
Chu y. Trong mdi ldi giai tren, ta nen lua chpn each danh sd cae nghiem sao cho "nghiem nay ga'p ddi nghiem kia' dupe the hidn bdi he thiic X2 = 2X|. Ne'u khdng lira chpn each danh sd eac nghiem nhu vay thi dieu kien "nghiem nay ga'p ddi nghiem kia" dupe didn ta bdi he thiic (X| -2x2)(x2 - 2 x , ) = 0.
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95
b) Dieu kien dd phuong trinh (2) cd nghiem la A2 = 49 - 8;t > 0. Vdi didu kien dd, gpi hai nghiem eiia (1) la X3 va X4. Theo didu kien cua dd bai, ta cd : X3 + X4 = 7 X3X4 = 2k
x | + X4 = 25. TiJf dd suy ra ^ = 6. Khi dd, (2) cd hai nghiem la X3 = 3 va X4 = 4. c) Didu kien dd hai phuong trinh ed nghiem la Aj > 0 va A2 > 0, tiic la 49 k< ~^- Vdi cung ki hieu nhu tren, theo de bai ta ed he : 8 Xj + X2 = 5 X1X2 ^
fc
X3 + X4 = 7 X3X4 = 2k 2xj = X3. Tii dd ta cd hai ket qua sau : • k = 0. Khi dd phuong trinh (1) cd hai nghiem la x^ = 0 va X2 = 5, phuong trinh (2) cd hai nghiem X3 = 0 va X4 = 7 (thoa man didu kien eua bai loan vi X3 = 2xi). • k = 6. Khi dd phuong trinh (1) cd hai nghiem x^ = 2 va X2 = 3, phuong trinh (2) cd hai nghiem X3 = 4 va X4 = 3 (thoa man didu kien ciia bai toan vi X3 = 2xi). 3.65. a) TiJf phuong trinh thii hai trong he ta nit y = x + 1 rdi th^ vao phuong 2 trinh thii nha't va thu gpn thi dupe phuong trinh bac hai 2x - 7x - 4 = 0. Phuong trtnh nay cho ta hai nghiem x = - — va x = 4. Tuong ling ta dupe hai nghiem eiia he phuong trinh da eho la ( - —; y ) va (4 ; 5).
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b) Ta cd (2x + 3y - 2)(x - 5y - 3) = 0 o 2x + 3y = 2 hoac x - 5y = 3. Do dd, he phuong trinh da eho tuong duong vdi (I)
fx - 5y = 3 {2x + 3y = 2 \ hoac (II) / [x-3y = 1 [ x - 3 y = 1.
Hai he nay cho ta hai nghiem eua he phuong trinh da eho la (1 ; 0) va ( - 2 ; -1). c) Day la he phuong trinh ddi xiing ddi vdi hai in. Do dd ta giai bing each dat « = x + y va v = xy. Khi dd ta thu dupe he phuong trinh in M va v |M^ [2H
- 6H + 2v + 9 = 0 - V + 6 = 0.
Ta giai he phuong trinh (III) bang phuong phap the'; ket qua la he nay vd nghiem nen he phuong trinh da eho vd nghiem. -
1
2
2
d) Tr^ tiUig ve hai phuong trinh eua he ta dupe 3(x - y ) = 7(x - y). Phuong trinh nay tuong duong vdi X - y = 0 hoac 3x + 3y - 7 = 0. Do dd, he phuong trinh da cho tuong duong vdi (iV)p-2>'=^^hoae(V)p-2>'=^^ [x-y = 0 • [3x + 3 y - 7 = 0. He (IV) ed hai nghiem (0 ; 0) va (-7 ; -7) ; he (V) vo nghiem. vay he phuong trinh da eho ed hai nghiem (0 ; 0) va (-7 ; -7). 3.66. a) Vdi a = 2 , ta cd he
\x^+y^=6
[ix + yf = 4. Dat M = X + y va V = xy, ta dupe he phuong trinh in la M va v : w^ - 2v = 6 u^=4. He nay cd hai nghiem (M ; v) = (2 ; -1) va iu ; v) = (-2 ; -1). Do dd he phuong trinh da cho tuong duong vdi ( ^ " ^ = ^ o a c ( ^ " ^ = -2 [xy = -1 [xy = - 1 -
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7-BTOSlO.NC-A
97
Giai hai he phuong trinh tren, ta dupe 4 nghiem eiia he phuong trinh da cho la
(1 + V2 ; 1 - V 2 ) , ( l - V2 ; 1 - V2), (-1 + V2 ; - l - V 2 ) , ( - l + V2 ; - l - V 2 ) . b) Gia sii (x ; y) = (XQ ; yo) la nghiem duy nha't eiia he. Do he phuong trinh da cho la he phuong trinh ddi xiing ddi vdi eac in nen nd ciing cd nghiem la ix ; y) = (yp ; XQ). TU" tinh duy nha't cua he ta suy ra XQ = yg. Dodd xo^ +y^=
2ia + 1) _
2xo = 2ia + 1)
—r
a = 0.
4x1 = 4
(xo+yof =4 Ngupe lai, ne'u (3 - 0 thi he trd thanh
x^ + y^ = 2 (X + y)2 = 4.
Tuy nhien, he nay cd nghiem khdng duy nha't (dd tha'y hai nghiem eua nd la (1 ; 1) va (-1 ; -1)). Vay khong cd gia tri nao ciia a thoa man didu kien eiia diu bai. 3.67. Phuong an (B). 3.68. Phuong an (A). 3.69. Phuong an (A). 3.70. Phuong an (A). 3.71. Phuong an (D). 3.72. (a) <^. ( 4 ) ;
(b) ^ (1) ;
(c) ^ (3).
3.73. (a) o (1) ;
(b) <^ ( 3 ) ;
(c) <^ (2).
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7-BTDSlO.NC-B
phucmg IV BAT DANG THl/C VA BAT PHaONG TRJNH
A. NHONG KIEN THQC CAN NHO 1. Tinh chat cua bat dang thurc l)a>bvab>c=>a>c. 2)a>boa
+ c>b + c.
3) Ne'u c> 0 thi a > b '^ ac> be. Ne'u c < 0 thi a > b <:> ac < be. Cdc he qua 4)a>bvac>d=>a a + c>b<:> a>b-
+ c>b + d. c.
5)a>b>0vac>d>0=^ac>bd. 6) a > ^ > 0 va n G N* => a" > &" l)a>b>0=>Ja>^fb S)a>b=>^>^2. Bat dang thiirc ve gia trj tuyet doi Ddi vdi hai sd a, b tuy y, ta cd ^
Id - \b\ <\a + b\< \a\ + \b\.
3. Bat dang thurc giCira trung binh cong va trung binh nhan 1) Vdi mpi (3 > 0 , /7 > 0 , t a e d a + b ^ i—r a + b r-r ^ > y/ab ; ^ = ^Jab <^a = b.
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99
2) Vdi mpi a>0, b>0,c
>0,taed
a + b + c ^ 3r-r- a + b + c ->/—— r > y/abc ; ~ = yjabc <::>a = b = c. Ap dung. 1) Ne'u hai sd duong cd t6ng khdng doi thi tich cua ehiing Idn nha't khi hai sd dd bang nhau. 2) Ne'u hai sd duong cd tich khdng ddi thi tdng ciia chiing nhd nha't khi hai sd dd bang nhau. 4. Bie'n doi tUdng duang cac bat phudng trinh
Cho ba't phuong trinh fix) < gix) cd tap xac dinh ®, y = hix) la mdt ham sd xac dinh tren y^. Khi dd, tren 2), ba't phuong trinh fix) < gix) tuong duong vdi mdi ba't phuong trinh 1) fix) + hix) < gix) + hix) ; 2) fix)hix) < gix)hix) ndu hix) > 0 vdi mpi x e S); 3) fix)hix) > gix)hix) ndu hix) < 0 vdi mpi x e 3). 5. Bat phUdng trinh va he bat phuong trinh bcic nhat mdt an
• Giai va bien luan ba't phuong trinh ax + b<0.
(1)
b 1) Ne'u a > 0 thi tap nghiem ciia (1) la S = - c o ; - ^ a 2) Ne'u a <0 thi tap nghiem ciia (1) la S = — ;+co V a 3) Ne'u a = 0 thi (1) trd thanh Ox < -b Do dd (1) vd nghiem (5 - 0 ) neu ^ > 0 ; (1) nghiem diing vdi mpi x (5 = R) ne'u ^ < 0. • Di giai mdt he ba't phuong trinh mdt in, ta giai tCmg ba't phuong trinh eiia he rdi la'y giao eiia cac tap nghiem thu dupe. booktoan.com 100
6. Dau cua nhj thiirc bac nhat 1) Bang xet da'u eiia nhi thiic bac nha't ax + b ia =^ 0) -co Trai da'u vdi a
ax + b
_b_ a 0
+00
Cung dau vdi a
2) Ne'u a > 0 thi \x\ < a <=> -a < X < a, X> a
X > o <=>
X < -a.
7. Bat phucfng trinh va he bat phuOng trinh bac nhat hai an 1) Cach xac dinh midn nghiem eua ax + by + c <0 ia^ + b^ ^ 0). (1) - Ve dudng thang id) : ax + by + c =^ 0 ; - La'y didm A/(xo ; yo) ^ id). Ne'u OXQ + by^j + C < 0 thi niia mat phang (khong ke bd id)) chiia didm M la midn nghiem ciia (1). Ne'u axQ + byQ + c > 0 thi niia mat phang (khdng kd bd id)) khdng ehda didm M la midn nghiem ciia (1). ^
9
9
Chii y. Ddi vdi bat phuong trinh ax + by + c <0 ia" + b i^ 0) thi each xac dinh midn nghiem cung tuong tu, nhung midn nghiem la niia mat phang kd ca bd. 2) Cach xac dinh mien nghiem ciia he ba't phuong trinh bac nha't hai an - Vdi mdi bat phuong trinh trong he, ta xae dinh midn nghiem eiia nd va gach bd midn edn lai. - Sau khi lam nhu tren \An lupt ddi vdi ta't ea cae bat phuong trinh trong he va tren ciing mpt mat phang toa dp, midn edn lai khdng hi gach chinh la midn nghiem eua he bat phuong trinh da cho. 8. Dau cua tam thurc bac hai 1) Cho tam thire bac hai fix) = ax^ + bx + c ia ^ 0).
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101
• Neu A < 0 thi fix)
ciing d^u vdi he sd a vdi mpi x e R, tiic la afix) > 0 vdi mpi x e R.
• Ne'u A = 0 thi fix)
cung da'u vdi he sd a vdi moi x ^ - - — . tde la 2a <^f{x) > 0 vdi moi x =^ -^r—, 2a cd hai nghiem phan biet x, va X2 (xj < X2). Khi
• Ne'u A > 0 thi fix) dd fix)
trai da'u vdi he sd a vdi mpi x nam trong khoang (x^ ; Xj) (tiic la
vdi Xj < X < X2) va fix)
ciing da'u vdi he sd a vdi mpi x nam ngoai doan
[xj ; X2] (tu"c la vdi x < X| hoac x > X2). Ndi each khae, « / U ) < 0 <=> X e (x^ ; -^2)'
afix) > 0 O
X >
XT
\a>0
J
2) \/x s R, ax + bx + c> 0 c^
X < Xj
[A
< 0.
a \a<0 Vx s R, ax'^ + bx + c < 0 <:> { A < 0.
B. DE BAI
§1. BAT D I N G THirc VA CHimC MINH BXT DANG THlfC 2
2
4.1. a) Chiing minh rang a + b - ab>0v6'i
mpi a, ft e R.
Khi nao ding thiic xay ra ? b) Chiing minh rang ne'u a> b thia
- b >ab - a b v6i mpi
4.2. Chiing minh rang a) a + b >ab + ab vdi mpi a, ft e R. b)ia + b + cf < 3ia^ + b^booktoan.com + c^) vdi mpi a, b, c e R. 102
a,be'B
4.3. Cho a, ft, c \a ba sd duong. Chiing minh rang . . XT--
; a) Nd^u a < ft thi - < ft b + c
,
1 ^ (^
a + c
b) Neu a > ft thi — > ft ft + c
a c 4.4. Cho «, ft, c, d la bdn sd duong va — < —. Chdng minh rang : , a +b c +d a) — r — < — —
, , (3 + ft f + J b) >
4.5. Cho ft, ti la hai sd duong va — < — Chihig minh rang a 'b^
a +c c b + d'^'d'
4.6. Cho a, ft, c, d la bdn sd duong. Chiing minh rang 1<
a b e d <2. + + a +b +c h +c +d c +d +a + d +a +b
4.7. Chiing minh rang
x" + I > 0 vdi mpi x > - 1 , « € N 4.8. Cho a, ft, c la sd do ba canh \ A,B,C mdt tam giac. Chiing minh rang :
la sd do (dp) ba gde tuong ling cua
a) ia - b)iA - 6) > 0 ; khi nao dang thiic xay ra ? 6Q0 < aA + bB + cC ^ g^o . ^j^. ^^^ ^^ a +b +c
^^^^ ^^ j.^ ^
iGai y. Su dung bat dang thUe tam giac). 4.9. a) Chiing minh ring, vdi mpi sd nguyen duong k ta deu cd 1
ik + \)4k
<2
1
1
V-[k ^/^7T
b) Ap dung. Chiing minh ring 1
1
1
1
— + —-j= + —1= + ... +
2
3V2
4V3
1=
<2.
( « + 1)V«
4.10. a) Cho /: > 0, ehiing minh — < -—- - - . i—
AT — 1 K booktoan.com
103
b) Ttr ket qua tren, hay suy ra 1 1 1 1 n — + — + — + ... + — < 2 . 1^ 2^ 3^ n^ 4.11. a) Cho hai sd a, bia^ ft). Tim gia tri nho nha!t ciia bidu thde fix) = (X - a)^ + (X - ft)2 b) Cho ba sd a, ft, c ddi mdt khae nhau. Tim gia tri nho nha't eua bidu thiic
gix) = ix-a)^ + ix-b)^ + ix-c)^. 4.12. Vdi cac sd a, ft, c tuy y, chiing minh cac ba't dang thiie sau va neu rd ding thiic xay ra khi nao ? a)|<3| + |ft| > |a - ft| ; b) \a + b + c\ < \a\ + IftI + |c| 4.13. Vdi cac sd a, ft, c tuy y, ehiing minh ba^t ding thiic |a - ft| + |ft - c| >
|G
- c|.
4.14. Tim gia tri nhd nha't cua bidu thde fix) = Ix - 2006] + Ix - 20071. 4.15. a) Chiing minh ring x + |x| > 0 vdi mpi x e R. b) Chiing minh ring vx + Vx"^ - x + 1 xac dinh vdi mpi x G R. 4.16. De chiing minh x(l - x) < — vdi mpi x, ban An da lam nhu sau : Ap dung ba^t ding thiic giUa trung binh edng va trung binh nhan eho hai sd X va 1 - X, ta cd ^Jx(l - X) <
X+ 1-X
2
1
2
Dodd
x(l-x)
4.17 Cho ba sd khdng am a, ft, c. Chiing minh cac ba't ding thiic sau va chi rd ding thiic xay ra khi nao : a) ia + b)iab + 1) > 4aft ;
b) (a + ft + c)iab + bc + ca) > 9abc.
4.18. Cho ba sd duong a, ft, c, ehiing minh ring :
-f
ftY.
1+ c
c^
1+ - >
4.19, Chiing minh ring : Ne'u 0 < a < ft thi a < ~
< Voft < -—— < ft.
4.20. Tim gia tri nho nha't ciia cac ham sd sau 1 2 vdi 0 < X < 1. b) gix) = — + X 1-x
a) fix) - x^ + 4 ;
4.21. Cho a>0, hay tim gia tri Idn nha't ciia y = xia-
2xf vdi 0 < x < y •
4.22. Cho mdt ta^m tdn hinh chu nhat ed ki'eh thude 80 em x 50 em. Hay cit di b bdn gde vudng nhiJng hinh vudng bing nhau dd khi gap lai theo mep cit thi dupe mpt cai hpp (khdng nip) ed thd tich Idn nha't. 4.23. Chiing minh ring a) Ne'u x^ + y^ = 1 thi |x + 2y| < Vs ; b) Ne'u 3x + 4y = 1 thi x^ + y^ > ~ 4.24. Cho a, ft, c la ba sd duong. Tim gia tri- nho nha't ciia A=
a b c ft + c + c + a + a + b
4.25. Trdn mat phing toa dp Oxy, ve dudng tron tam 0 cd ban ki'nh R iR> 0). Tren cae tia Ox va Oy l^n lupt la'y hai didm A va B sao cho dudng thing AB ludn tie'p xiic vdi dudng tron dd. 5 dd tam giac OAB cd dien tich nhd nha't. Hay xae dinh toa dp eiia A va booktoan.com 105
§2. DAI C i r O N G Vfi B A T PHLfONG TRINH 4.26. Trong eac menh dd sau, menh dd nao diing, menh dd nao sai, vi sao ? a) 2 la mdt nghiem ciia ba't phuong trinh x^ + x + 1 > 0. b) - 3 khdng la nghiem cua ba't phuong trinh x^ - 3x - 1 < 0. c) a la mdt nghiem ciia ba't phuong trinh x + (1 + a)x - a + 2 < 0. 4.27. Cac cap ba't phuong trinh sau cd tuong duong khdng, vi sao ? a) 2x - 1 > 0 va 2x - 1 + x-2
> X- 2 '
b) 2x - 1 > 0 va 2x - 1 + ——- > x+2 x+2 ' c) X - 3 < 0 va x^(x - 3) < 0 ;
d) x - 3 > 0 va x^(x - 3) > 0 ;
e) X - 2 > 0 va (x - 2)^ > 0 ; g) x - 5 > 0 va (x - 5)(x^ - 2x + 2) > 0. 4.28, Tim didu kien xac dinh roi suy ra tap nghiem cua mdi bat phuong trinh sau : a) V x - 2 > V 2 - X ;
b) V2x - 3 < 1 + V2x - 3 ;
e)
d)3x+^—>2+ x-2
. -^ six-3
< .^ ; Vx-3 '
^ x-2
4.29. Khong giai bat phuong trinh hay giai thi'eh tai sao cac bat phuong trinh sau vo nghiem : a) V 7 ^ 2 + 1 < 0 ; b) (x - 1)^ + x^ < -3 ; e) x^ + (x - 3)^ + 2 > (x - 3)^ + X- + 5 ; d) Vl + 2(x + l)^ + > / l 0 - 6 x + x^ < 2 . 4.30. Khong giai bat phuong trinh, hay giai thich tai sao cae ba't phuong trinh sau nghiem diing vdi mpi x : a) ,v* + x^ + 1 > 0 ;
b) ^^^f^ > 0 ; x^ + 1
c) X^ + (X - 1)^ + — > x^ X- +booktoan.com 1
106
4.31. Tim didu kien xdc dinh ciia cac ba't phuong trinh sau : 1
1
^
'
• a) (X +1)2T + X - n 3 > 2 ;'
^^ v G m 1 1 b) VTTT , + (X - — — > 2)(x - 3) x-4
4.32. Di giai baj: phuong trinh Vx - 2 > V2x - 3 (1), ban Nam da lam nhu sau : Do hai ve' eiia ba^t phuong trinh (1) luon khong am nen (1) tuong duong vdi (Vx-2)2 > (V2x - 3)2 hay X - 2 > 2x - 3. Do dd X < 1. vay tap nghiem ciia (1) la (-QO, 1). Theo em, ban Nam giai da dung chua, vi sao ? 4.33. Ban Minh giai ba't phuong trinh
, < (1) nhu sau : Vx2-2x-3 '^ + 5
(l)<=>x + 5< Vx^ - 2x - 3 <=> (X + 5)2 < x^ - 2x - 3 <=> 12x + 28<0<=>x< - - . Theo em, ban Minh giai diing hay sai, vi sao ?
§3. BAT PHLTcnSIG TRINH VA Hfi BAT PHLfONG TRINH B A C N H A T M O T
XN
4.34. Giai cac ba't phuong trinh sau va bidu didn tap nghiem tren true sd : a)2(x-l)+x> ^ ^ + 3 ;
b) ix + ^f
e)x(7-x) + 6 ( x - l ) < x ( 2 - x ) ;
d ) ^
+^
< (x - ^2)^ + 2 ; +^ > 3 + |-
4.35. Giai eac ba't phuong trinh a) (x + 2)Vx + 3Vx + 4 < 0 ;
b) (x + 2)V(x + 3)(x + 4) < 0 ;
c) yjix - 1)2(X - 2) > 0 ;
d) ^2x - 8 - V 4 x - 2 l > 0.
booktoan.com 107
4.36. Giai cac he bit phuong trinh sau va bidu didn tap nghiem tren true sd : 3x + - < X + 2 a)
6x-3
b) < 2x + 1;
4x + 5 < 6 2x + 3 >
X -
3
7x - 4
4.37. Giai va bien luan cac bat phuong trinh (in x): a) mix -m)>0
;
b)(x- l)m>x + 2;
^ X - ab X - ac x - be +
d) ftx + ft < a
-ax.
Vx2 - 1 - Vx + 1 > X + 1
(1)
nhu sau : (X - 1 > 0 ^.. _ ] x 2 - l > 0 _ [(x-l)(x + l ) > 0 Dieu kien : < o <=> -^ <=>x> 1. x + l>0 x + l>0 [x + l > 0 Khi dd ba't phuong trinh (1) ed dang V(.v-l)(x + l ) - V x + l > x + l . Chia hai ve eho Vx + 1 > 0 , ta ed Vx - 1 - 1 > Vx + 1 . Vi X > 1 nen Vx - 1 < Vx + 1, do dd V x - l - 1 < Vx + 1. vay ba't phuong trinh (1) vd nghiem. Theo em, ban Nam giai diing hay sai, vi sao ? 4.39. Tim cac gia tri ciia m dd he bit phuong trinh sau ed nghiem : Ix + 4m^ < 2mx + 1 l3x + 2 > 2 x - I . 4.40. Tim cae gia tri eua m de he bat phuong trtnh sau vd nghiem : \mx + 9 <3x + m^ 4x + 1 < -X + 6. booktoan.com 108
-..^
§4. DAU CUA NHI THirc BAC
NHAT
4.41. Xet da'u cua cac bidu thiic sau bing each lap bang : 2-3x 5x-l '
a)(3x-l)(x + 2);
b)
c ) ( - x + l ) ( x + 2)(3x+ 1);
d) 2 -
2+x 3x-2
4.42. Phan tfeh eac da thiic sau thanh nhan tii rdi xet da'u mdi da thiic a'y : a) 9x - 1 ;
b) -x^ + 7x - 6 ;
c) x^ + x2 - 5x + 3 ;
d)x2 - X - 2 V 2
4.43. Xet d^u cac bidu thiic sau : a)
c)
1 3- X
1
b)
3+ X '
x2 - 6x + 8 x2 + 8x - 9
X + 4x + 4
d)
x' - 2x2
|x + l | - T x2 + X + 1
4.44. Giai cac ba't phiiong trinh sau ; a) (-V2x + 2)(x + l)(2x - 3) > 0 ;
3x + 1
4.45. Giai eac phuong trinh sau : a) |5 + x| + |x - 3| = 8 ;
b) X - 5x + 6 == x'^ - 5x + 6 ;
c) |2x - l| - X + 2 ;
d) Ix + 2I + Ix - l| = 5.
4.46. Giai cdc bat phuong trinh sau : a) |3x - 5| < 2 ; c) Ix - 2| > 2x - 3 ;
b)
2-x
x +1
>2;
d) Ix + l| < 1x1 - X + 2
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109
§5. BAT PHUONG TRINH VA H£ BAT PHUONG
TRINH BAC NHAT HAI XN
4.47. Xae dinh midn nghiem eiia eac ba't phuong trinh sau (x, y la hai in) : a)2(x + y + l ) > x + 2 ;
b) 2(y+ x) < 3(x+ 1)+ 1 ;
c)y + 0 . x > 5 ;
d)0.y + x < 3 .
4.48. Xae dinh midn nghiem eiia cac he ba't phuong trinh sau : ' [x + 3y > - 2 ;
[y < 3.
4.49. Xac dinh midn nghiem eiia eac he ba't phuong trinh sau : y >0
X - 3y < 0
a)
X + 2y > - 3 y+ X< 2;
X
y
13
2
1
4,50. Xac dinh midn nghiem eua he ba't phuong trinh | | x - l| < 1 [\y + 1| < 2. 4.51, a) Xae dinh midn nghiem eua he bat phuong trinh 0 < X< 5 0 < y < 10
b) Tim gia tri nho nhat eua bidu thiic 7 = 2x - 2y + 3 tren midn nghiem Of cau a, biet ring midn nghiem do la mien da giac va T ed gia tri nhd nha't tai mpt trong eac dinh ciia da giac dd. 4.52, Mdt XI nghiep san xuat hai loai san phim ki hieu la / va //. Mpt tin san phim / lai 2 trieu ddng, mpt ta'n san phim // lai 1,6 trieu ddng. Mudn san
booktoan.com 110
xua't 1 ta'n san phim / phai dung may Mj trong 3 gid va may M2 trong 1 gid. Mudn san xua't 1 tin san pham // phai dung may M^ trong 1 gid va may M2 trong 1 gid. Bidt ring mpt may khong the diing di san xua't ddng thdi hai loai san pham ; may Mj lam viec khong qua 6 gid trong mpt ngay, may M2 mdt ngay chi lam viec khong qua 4 gid. Gia sir xi nghiep san xua't trong mOt ngay dupe x (tin) san phim / va y (ta'n) san phim //. a) Viet cic ba't phuong trinh bidu thi cac didu kien cua bai toan thanh mpt he bat phuong trinh rdi xae dinh midn nghiem (5) eiia he dd. b) Gpi T (trieu ddng) la sd tidn lai mdi ngay eiia xi nghiep. Hay bidu didn T theo X, y. e) O cau a) ta thay (5) la mpt mien da giac. Bidt ring T ed gia tri Idn nhat tai (XQ ; yo) vdi (XQ ; yo) la toa dp ciia mpt trong cae dinh cua (5). Hay dat ke' hoach san xuit ciia xi nghiep sao cho tdng sd tidn lai cao nhit.
§6. D A U C U A TAM THU'C B A C HAI 4.53. Xet da'u eda cic tam thiJc bac hai : a) 2x2 + 2x + 5 ;
^^ _^2 + 5^ _ 5 .
e) 2x2 ^ 2x42 + 1 ;
d) -4x2 _ 4^ ^ j .
e)V3x2 + (V3+ l)x+ 1 ;
f)x^ + i45-
g)-0,3x2 + x - 1,5;
h ) x 2 - ( V 7 - l)x + V3.
4.54. Xet da'u eiia cac bidu thiie : x-7 4x^• - 1 9 X + 12 '
a)
3x-2
.,
b)
d)
x^ - 3x2 + 2 '
c) x2 - 3 x - 2 .
f)
-X" + X - 1
Vs ;
llx + 3 - x 2 + 5x --7
x2 + 4x -- 1 2
V6x2
+ 3 x + V2 '
X^ - 5 x
/ -
booktoan.com
e)
\)x-
+4
4x^ + 8 x - 5
X • - 4X- + 8X - 3
111
4.55. Chiing minh ring cac phuong trinh sau ludn ed nghiem vdi mpi gii tri ciia tham sd m : 2
1
?
a)x + ( m + l ) x + m - - - 0 ; 9
3
b)x^-2(/?i - l)x +m - 3 = 0 ; -7
1
c) X + (m + 2)x + - m + - = 0 ;
•
d) (m - l)x'^ + (3m - 2)x + 3 - 2m = 0.
4.56. Chiing minh ring cac phuong trinh sau vd nghiem dii m liy bit ki gii tri nao: a)(2m^ + l)x^-4mx + 2 - 0 ; b) - x ^ + (m + l)x + m^ + m + 1 = 0 ; c) X + 2(m - 3)x + 2w^ - 7m + 10 = 0 ; d ) x 2 - ( V 3 m - l)x + m2- V3m + 2 = 0. 4.57. Tim cic gia tri eua m di mdi bidu thiic sau lu6n duong : a) x^ - 4x + m - 5 ;
b) x^ - (m + 2)x + 8m + 1 ;
c) x^ + 4x + (m - 2f ; d) (3m + l)x^ - (3m + l)x + m + 4. 4.58. Tim cac gia tri eua m di mdi bidu thde sau ludn am : a) (m - 4)x2 + (m + l)x + 2m -1 ;
b) (m + 2)x^ + 5x - 4 ;
c) mx - 12x - 5 ;*
d) -x^ + 4(m + l)x + 1 - m^
§7. BAT PHUONG TRINH BAC HAI 4.59. Giai eac bit phuong trinh : a) 2x2 - 7 X - 15 > 0 ; c) x(x + 5) < 2(x2 + 2) ;
^^ ^^x^ _ ^^^ _ io5 < 0 ; d) 2(x + 2)^ - 3,5 > 2x ;
1 2 e) -X - 3x + 6< 0. 4.60. Giai cac bit phuong trinh : 2x - 5 1 . , x2 - 5x + 6 ^ X + 1 a) -^ < ; b) — > ; x2 - 6x - 7 x-3 x2 + 5x + 6 x 2 1 ^ 2x - 1 ^,2 1 1 ^n c) — > booktoan.com — ; d) - + r < 0. x^^x + \ x + l :,^ + i X x-1 x +1 112
4.61. Tim cac gia tri nguyen khdng am ciia x thoa man bit phuong trinh : 2x
x +3
x+2
x2-4
2x-x'
4.62. Giai cac bit phuong trinh : a)(x-l)Vx2 - x - 2 > 0 ;
b)
f-
X + X+ 6
2x + 5
>
r-
X + X+ 6
x-4
4.63. Giai eac he bit phuong trinh va bidu didn tap nghiem ciia ehiing tren true sd: a)
x'^ - 2x - 3 > 0
x2-i>0
b)
x2-llx + 28>0; c)
-2x^ + 5x - 3 > 0 ;
3x^ - 4x + 1 > 0
x'' - 8x + 7 < 0
d)
3x2 - 5 x + 2 < 0 ;
x2 - 8x + 20 > 0.
4.64. Giai cae he bit phuong trinh va bidu didn tap nghiem cua ehiing tren true sd: x^ - 4x - 5 < 0
x^ - 1 2 x - 6 4 < 0 b) x2 - 8x + 15 > 0 3 ^ ^13 — < X < —• 4 2
a) x2 - 6x + 8 > 0 2x - 3 > 0 ; 4.65. Tim tap xic dinh ciia ham sd sau :
fix)=^
3-3x
-x^ - 2x + 15
-I.
4.66. Tim cic gia tri ciia tham sd m dd he bit phuong trinh : Jx2 -3x-4<0 , ... a) { c6 nghiem ; (m-l)x-2>0
^, x'^ +10X + 16 < 0 , ^.. b) < v6 nghiem. mx > 3m + 1
4.67. Tim cdc gii tri cua tham sd m di mdi phuong trinh sau ed nghidm : a) 2x^ + 2(m + 2)x + 3 + 4m + m2 = 0 ; b) (m - l)x2- 2(m + 3)x - m + 2 = 0.
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113
4.68. Tim cac gia tri ciia tham sd m dd mdi bit phuong trinh sau nghiem dung mpi gia tri x : a ) ( m + l ) x 2 - 2 ( m - l)x + 3 m - 3 > 0 ; b) (m2 + 4m - 5)x^ - 2(m - l)x + 2 < 0 ; x2 - 8x + 20 < 0; e) — ^ mx + 2(m + l)x + 9m + 4 3x2 _ 5 , 4 d) ^^ ^-^-> 0. (m - 4)x + (1 + m)x + 2m - 1 4.69. Tim eac gia tri eiia m dd phuong trinh : a) X + 2(m + l)x + 9m - 5 - 0 cd hai nghiem am phan biet; b) (m-2)x - 2mx + /?/ + 3 = 0 ed hai nghiem duong phan biet. 4.70. Cho phuong trinh : (m - 2)x'^ - 2(m + l)x2 + 2m - 1 = 0. Tim eac gia tri ciia tham sd m dd phuong trinh tren cd : a) Mdt nghiem ; b) Hai nghiem phan biet; e) Bdn nghiem phan biet.
§8 MOT S6 PHUONG TRINH VA BAT PHUONG TRINH QUY vt BAC HAI 4.71. Giai cac phuong trinh : a) 9x + V 3 x - 2 = 10 ;
b) V-x^ + 2x + 4 = x - 2 ;
c) Vx2 - 2x - 3 = 2x + 3 ;
d) V9 - 5x = V T ^ +
-^^ V3^
4.72. Giai eac phuong trinh sau : a) (x + l)Vl6x + 17 = (x + l)(8x - 23) ; b) - r — ^ x2 - 4x + 10 2x
x2 + 4x - 6 = 0 ; 13x
,
e) —z + —. =6 ; 2x2 _ 5 ^ . + 3 2x2 +booktoan.com X+3 114
J,-. 2
d) x-^ +
(
X
\
= l.
Vx-i; 8-BTOSlO.NC-B
4.73. Giai cac phuong trinh sau : a) 2 x ^ - 3 - 5 ^ 2 x 2 + 3 = 0 ;
b) 2x^ + 3x + 3 - 5 ^2x2 + 3x + 9 ;
c)9 -VsT^ 7x^ =
d)x2 + 3 - V2x2 - 3x + 2 = I (x+1).
2 '
4.74. Tim ta't ca cac gia tri x thoa man : 1
1
1
^
a) |x + X - 11 = 2x - 1 va X < —- ; b) 1x2 + 2x - 4I + 2x + 6 = 0 va X + Vis < 1 ; c) Ix + 3| + x2 + 3x = 0 ; d) 1x2 - 20x - 9 U I3x2 + lOx + 21I. 4.75. Giai cac phuong trinh sau : a) x2- |2x - l| = 0 ;
b) 1x2 - 2x - 3I = x2 - 2x + 5
c)|2x-3|=|x-l| ; 4.76. Giai cac phuong trinh sau :
d) jx^ - 2x - 3| = 2.
a) ylx + 3~ 4 V x ^ + Vx + 8 - 6 V x ^ = 1 ;
b) V77Vl47^^ + ^Ix - Vl4x - 49 = Vl4 ; c) I2V2IXI-I - 1 | = 3 ; d) X + V l - x ^ l = -V2(2x2 - 1). 4.77. Giai cae bit phuong trinh sau : a) V-x2 - 8 x - 1 2 > x + 4 ; c)
V2-X + 4x - 3
>2 ;
b) ^5x2 + 61x < 4x + 2 ;
4.3x^
-3
4.78. Giai cae bit phuong trinh sau
4~- + 6x - 5 > 8 - 2x b) V-x^
a) Vx + 3 < 1 - X ; c)4\x + l-]> V5x2+61x ; V
2J
'
d) yjix^ - x)^
>x-2.
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4.79. Giai eac bit phuong trinh : a) |3 - Vx + 5| > X ;
b) 7|4 - Vx + 9| > x - 9 ;
c) x + 13 + | 2 4 - 6 V 6 - x | > 0 ; d) ^xix + 6)+ 9 ~ yjx^ -6x+ 9 > 1. 4.80. Giai cac bit phuong trinh sau : a)(x^ + x+ l)(x^ + x + 3)> 15 ; b) (x + 4)(x + 1) - 3 Vx2 + 5x + 2 < 6 ; c) x2 - 4x ~ 6 > ^2x2 - 8 x + 12. 4.81, Giai cae bit phuong trinh sau : a) (x - 3)Vx2 + 4 < x2 - 9 ;
b) /
< 3x + 2 .
V5x2 _ ^ 4.82. Dd'i vdi mdi gia tri ciia tham sd m, hay xae dinh sd nghiem ciia phuong trinh ; V2|x| - x = m.
BAI TAP ON TAP CHUONG IV 4.83. Khong diing may tinh va bang sd, hay so sanh . 3 - Vr23 . 2 - V37 a) ;;; va ; 4
. 3V7 + 5V2 , ^ _ b) ^ va 6,9.
3
VS
4.84. Chiing minh ring ne'u \a\ <\,\b-
l|< 10, |a - c|< 10 thi \ab - c\ < 20.
4.85. Cho cae sd khdng am a, b, c. Chutig minh rang : a^ +b^
2,3
1.
a) — - — >3ab -16 ; h)a + b + 2a^ + 2b^ > 2ab + 2h yfa + 2a yfb , 4.86. Tim gia tri nho nha't ciia cac bidu thire : a)A = a^ + b^ + ab-3a~3b + 2006 ; h)B = a^ + 2^2 _ 2ab + 2a-4b- 12.
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116
4.87. Chiing minh ring ndu eac sd a, b, c deu duong thi : a)ia + b + c)ia- + \? + c-) > 9abc ; ,
a"
b'
c'
c) b+c + c+a + a+b
>
h)—+ a
a +b +c ab > 2 a+b
^ + ^>a b c be + -b+c
+ b + c;
ca + c+a
4.88. Hay xae dinh gia tri nho nhit cua cac bidu thde sau : 3,)P= |;(: + i| + |2;c + 5| + | 3 x - 1 8 | ; b) 2 = |x - l| + |y - 2| + |z - 3| vdfi |x| + \y\ + \z\ = 2006. 4.89. Giai cae bit phuong trinh sau : , 3x - 1 a) —r=
^
^
x + 2>2x-3;
. , 2x + 5 b) —
^ ^ 3x ~ 7 3 < — ^ +X + 2 ;
V3 c ) ( l + V 3 ) x < 4 + 2V3 ;
d ) ( x - V 5 ) 2 > ( x + V 5 ) ^ - 10.
4.90. Giai va bien luan cae bit phuong trinh sau theo tham sd m : a) mx - 1 > 3x + m ; e)
3x
(m -- 7)2
<
x-1 m-1 '
e) mx2 + 4x + 1 < 0 ;
b) m(m-2)x + 1 > m - I ; d) X + 2nvc + 5 > 0 ; f) (m - 3) x^ - 2(m + I )x - (2m - 3) < 0.
4.91. Tim tit ca cae nghiem nguyen ciia mdi he bat phuong trinh sau : 42x + 5 > 28x + 49 a)
oX + J
-.
^^
45x - 2 > 6x + b)
— - — < 2x + 25 ;
2(3x - 4) <
9x-l4
4.92. Xac dinh cae gia tri ciia m dd mdi he bit phuong trinh sau ed nghiem : ' 7 x - 2 > - 4 x + l9 a)
2x - 3m + 2 < 0 ;
b)
V2 X + 1 > X - V2 m + X > 2.
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4.93. Giai eac bit phuong trinh sau : a) |x - l| + |x + 2| < 3 ; c)
|2x - l|
b) 2|x - 3| - |3x + l| < x + 5 ;
1 — •
<
x2 - 3x - 4
2
4.94. Giai eac bit phuong trinh sau : a) (x^ + 3x + l)(x^ + 3x - 3) > 5 ; c)
20
x 2 - 7 x + 12
+
r + 1 >0; -^-4
b)(x2-x-l)(x2-x-7)<-5; d) 2x'^ + 2x -
15 x'^ +X + 1
+ 1 <0.
4.95. Tim cac gia tri eiia x thoa man he bit phuong trinh : 3x2 + ^ ; ^ . _ 4 < o
2x2 + 9x + 9 > 0
b)
a)
x2 - 8x - 20 < 0 ;
5x2 - 7x - 3 < 0 ;
3x'^ - 7x +
2(x - 1) - 3(x - 4) > X + 5 c)
3x - 4 •^•^ ^
X + 4x + 4
d) >0;
>1
x2+l
3x2
_j^^ <2.
x2+l 4.96. Xac dinh cae gia tri eua tham sd m di mdi bit phuong trinh sau nghiem dung vdi mpi x. a)^ _X_ + mx - 1 < 1 ;
b)-4<
2x2 _ 2x + 3
2x + mx - 4
<6.
-x2 + X - 1
4.97. Tuy theo gia tri eiia tham sd m, hay bien luan sd nghiem phuong trinh (m + 3)x'^ - (2m - l)x^ - 3 = 0. 4.98. Xet da'u eac bidu thdc sau : 7x - 4
b)
x^ - 5x + 4 , x2 + 5x + 4
c)
6x2 - X + 5
118
x^ - 17x2 ^ gQ
15x^ - 7 x - 2
d)
booktoan.com
x(x2 - 8x + 5)
4.99. Giai eac bit phuong trinh : a) : ^ / ^ ^ + V 7 ^ > 5_ . Vx - 3 Vx - 3
b)Vx«-4;.3^4>x-^;
c) V3x2 + 5x + 7 - ^3x2 + 5x + 2 > 1. 4.100. Giai eac bit phuong trinh : a) V x - l - V x - 2 > V x - 3 ;
b) 2x(x - 1) + 1 > 4x^ -x + \ ;
4.101. Tim cac gia tri x thoa man : a) 1x2 - 2x - 3| - 2 > |2x - l| ;
b) 2|x + l| < |x - 2| + 3x + 1 ;
c) | V x - 3 - l| + |Vx + 5 - l| > 2 ;
d) jx - 6| > |x2 - 5x + 9|.
4.102. Giai cac bit phuong trinh sau : a)
, < 3; x-3
.. |x + 2|-|x| b) —, >0 ;
3x + l
V^
c) I ^1 > Ix + 2| ; |x + 3| - 1 4.103. Cho phuong trinh im-^)x^ m thi
d) I ^r—- > Ix - 2|. |x - 51 - 3 - 3mx + m + 1 = 0. Vdi cac gia tri nao eiia
a) Phuong trinh da cho cd nghiem ? b) Phuong trinh da eho cd hai nghiem trai da'u nhau. 4.104. Tuy thude vao gia tri eua tham sd m, hay xac dinh sd nghiem eua phuong trinh : 1x2 - 2x - 3I = m. 4.105. Tim tit ea cae gia tri eua m de ling vdi mdi gia tri dd phuong trinh |l - mxl = 1 + (I - 2m)x + mx2 chi ed diing mpt nghiem.
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Gidi THifiU MOT sd cAu HOI TRAC NGHlfiM KHACH QUAN 4.106. Trong cac khang dinh sau day, khing dinh nao diing ? {aa + c
b)-^ \c < d
=>a~c
^ ac \c < d
., k < & \c < d
a c
2
2
1 b
h)a>b => yfa > yfb ;
g) a> b=> ac> be ; i)a + b>2
1 a
f)a>b
e) a> b => a > b ;
b d
{a>l b>\;
\a>l U'>1.
k)ab>
Chon phuang dn trd ldi ma em cho la diing d cdc bdi sau (ttr 4.107 ddn 4.114) 4.107. X = - 3 thupc tap nghiem ciia bit phuong trinh (A)(x+3)(x + 2 ) > 0 ; (C)x +
VT x^ > 0
(B)(x + 3)'(x + 2 ) < 0 ; 1 2 (D)^-!—+ , ^ ^>0. 1 + x 3 + 2x
;
4.108. Bit phuong trinh (x - \)^jxix + 2) > 0 tuong duong vdi bit phuong trtnh ( A ) ( x - l)V^Vx + 2 > 0 ;
(B) V u - l)2x(x +2)> 0 ;
^^^(x-l)V4x + 2 ) ^ ^ .
^P^(x-l)Vx(x + 2)^Q_
{x + 3y
(x-2)
4.109. Bit phuong trinh mx > 3 vO nghiem khi (A) m = 0 ;
(B) m > 0 ;
(C) m < 0 ;
(D) m # 0.
4.110. Bit phuong trinh 2 - x > 0 ed tap nghiem la 2x + l (A)|-2;2|; 120
(B)
(C)
-2-^\ booktoan.com
-2^^J=
^^M-i = 2
4.111. He bit phuong trinh l cd tap nghiem la [2x + 1 > X - 2 (A) (-«); - 3 ) ; (B) (-3 ; 2); (C) (2 ; +oo); (D) (-3 ; +co). 4.112. He bit phuong trinh ^-^ "^ ^^ ^ ^ cd nghiem khi [x < m - 1 (A) m < 5 ; (B) m > -2 ; (C) m = 5 ; (D) m > 5. 4.113. He bit phuong trtnh < ~ cd nghiem khi x-m > 0 (A) m > 1 ;
(B) m = 1 ;
4.114. He bit phuong trtnh
(C) m < 1 ;
x'^ - 4x + 3 > 0
(D)
m^l.
cd tap nghiem la
x2 - 6x + 8 > 0 (A) (-co ; 1) u (3 ; +co);
(B) (-oo ; 1) u (4 ; +co);
(C) (-00 ; 2) u (3 ;+«));
(D) (1 ; 4).
4.115. Hay ghep mdi ddng d cpt trai vdi mpt dong 6 cpt phai trong bang sau di dupe mdt khang dinh diing : a) x2 - 5x + 6 > 0 <^
(l)2
b) x2 - 5 x + 6 S 0 o
(2) X > 3 hoac x < 2
c) x2 - 5 x + 6 < 0 o
(3)2
d) x2 - 5x + 6 > 0 <=>
(4) X > 3 hoac x < 2 (5) 2 < X < 3
4.116. Didn diu (>, >, <, <) thi'eh hpp vao 6 trdng. Cho tam thiJe fix) = x^ + 2mx + m^ -m + 2 imVa tham sd'). a) fix) > 0 vdi mpi X G R khi
mn 2;
b) fix) > 0 vdi mpi x e IR khi
"i D 2 ;
c) Tdn tai x dd /(x) < 0 khi
mQ 2;
d) Tdn tai x dd fix) < 0 khi
m Q 2.
booktoan.com 121
C. HUONG DAN - LOI GIAI - DAP SO 4.1. a)aUb^-ab
= \a-^^
+ ^ > Ovdimpia, fo G M.
a--\ Diu bang xay ra khi va chi khi
-0 hay a - 6 = 0.
3b' =0 I 4 b) a^ -b^ - iab^ - a^h)= aia^ - b^) + bia^ - b^) - (fl + b)ia^ - b^)
= ia- b)ia + bf Do a>b
nen ia - b)ia + b)^ > 0, ta cd didu phai chiing minh.
4.2. a) fl^ + />"* - a^b - ab^ = a^ia - b) + b^ib - a) = ia - b)ia^ - b^) = {a-bfia^ (Vi a2 +^2 ^ ^ ^ ^ f
^M
+b^
+ab)>0.
+ ^ > o v a ( a - & ) 2 > 0 vdimpia,^^ (1)
b)(a + 6 + c)2 <3(a2 +^2 +^2^ <=> ^2 + ^2 ^ ^2 ^ 2a/> + 2ac + 2bc < 3a^ + 3b^ + 3c^ 0 ^ 2 + 6 2 + (-.2 ^ ab- ac ~ be > 0 o ( a - b)^ +ib-
cf + ic -af
>Q.
Bit ding thiic (2) ludn diing nen bit dang thiic (1) dupe chiing minh. 4.3. Tacd
a +c b+c
a b
cib - a) bib + c)
a) Ndu 0 < a < ^ va c > 0 thi ^ ^ — — > 0. Suy ra -^ < ^—bib + c) ^ b b+c cib -a) a a +c b) Neu « > /7 > 0 va r > 0 thi -^^ ~ < 0. Suy ra - > bih + c) ' b h +c
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(2)
4.4. a) Tu - < — suy ra - + 1 < — + 1, tuc la —;— < — ; — b d b d b d b) Tir — < — va a, b, c, d \a bdn sd duong nen — > —. suy ra b d a c b , d , , ,^ b + a d +c — + 1 > — + 1, tuc la > a c a c a c 4.5. Tii — < — va /?, 6^ la hai sd duong, suy ra ad < be hay ad - be < 0 ; be - ad> 0. T
' ^ + '^ _ ^ _ ^^ ~ ^ ^ ^"^^ b + d b~ ib + d)b^
,.^
a b
a +c b +d
f^. q + c _c _ ad-bc ' b + d 'd~ ib + d)d ^
c d
vay - < - — - < - . 4.6. Do a, b, c, d la eac sd duong nen
a +b +c
>
a +b +c +d
b b > b +c +d a +b +c +d c c > c +d +a a +b +c +d d d > d +a +b a +b +c + d Cpng v^ vdi ve' ciia cae bit dang thiic tren, ta suy ra a a+b+c , . , L a i CO
nen
b + ~ b+c+d
a : a+b+c
a ; a+b+c
<
+
a a+c
c
d
- + c+d+a•
c ;'c+d+a:
,
+ -:d+a+b > 1. <
c a+c
c :; < 1. c+d+a
Tuong tu -; r+ 7- < 1. Til dd suy ra ^ ' b +c +d d +a +b ^ a b c d + -, 7 + ; + -. r < 2. a+b+c b+c+d c+d+a d+a+b
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4.7. Ne'u X > 0 thi x" + 1 > 1 > 0. Neu - 1 < X < 0 thi Ixl < 1 suy ra Ixl" < 1 hay Ix"! < 1. Ttr dd ta cd -x" < 1 (vi -x" < lx"l). Vi vay x" + 1 > 0. 4.8. a) Ap dung mdi lien he giUa canh va gde trong tam giac, ta eo ; Neu
a>bth\A>B;
Ndu a< b thi
A
Vi vay ludn ed ia - b)iA - B) > 0, ding thUc xay ra khi va chi khi a = biA=B),
tiic la tam giac ABC can tai C.
b) • Theo eau a) ta cd ia - b)iA -B) + ib- c)iB - C) + ic - a)iC ~ A) > 0 <^ aA + bB + cC - bA - aB + bB - cB - bC + cC - aC - cA + aA >0 <=> 3iaA + bB + cC) - ia + b + c)iA + B + C)>0
o
aA + bB + cC . A + B + C = 60° a+b+c 3
Ding thiie xay ra khi va chi khi A ~ B = C, tvtc \a tam giac ABC la tam giac ddu. • Lai cd
a + b>c;
b + c>a;c
+ a>b nen
aA + bB + cC
2iaA + bB + cC)
aA + bB + cC <9Qf. a +b +c
Tir dd suy ra 4.9. a) Ta cd
+ B + C)ia + b + c).
1
Tk
ik + \)y[k
(^ + 1)^
= V^
^41-1^]
4k + ^k + \K4k
f 1+
^k + \
Ji ylk + \X4k
^
V^ + i J " \4k
booktoan.com
124
< 2
VT+T
'j 1_^J \ 1 1_ 1 b) - + — = + — = + . . . + <2 2 3 ^ ^ 4V3 "' in + \)4^ . V2 V2 V3 V3 V4'^'""^ 1
1 1 = 2 1<2. yjn + 1 V« + i.
+ 4n
1 . 1 /: _ 1 ^
4.10. a) Vdi )t > I ta cd : 4/:3- ^ 4^2- < (/: _^ 1)^ UN 1
1
1
1
, ,
1
1
1
1
1
1
1
= 2-i<2. n 4.11. a) /(x) = (x - af + (x - 6)2 = 2x2 - ^^^ ^ ^^^ + ^2 ^ ^2 = 2 (a - by
Ta cd fix) > ^--^J ia-b)^ —
X -
a + b\
+
ia-b)
vdi mpi a, b ; ding thiie xay ra khi
=0, tiic la X =^^
vay fix) dat gia tri nho nha't la
^. a+b taix= —r
Chi4 y. Tranh sai lim khi suy luan ring (x - a)^ + (x - 6)2 > 0 vdi mpi x nen gia trj nho nhit eua fix) la 0. b) Hudng ddn. Vie't gix) dudi dang J
X
-
a + b + c\^
4.12. a) Iflj + |6| = \a\ + \-b\
+
ia- bf +{b- cf + (c - af
>\a-b\.
Dang thiic xay ra khi va ehi khi ab < 0. b) Hudng ddn. < | 3 + 6 + cj < jo + 6| + |c|. 4.13. U - /?| + I/) - d > Ifl - 6 + 6 - cl = Ifl - cl.
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4.14. fix)
= \x- 20061 + Ix - 20071 > |x - 2006 - (x - 2007)| = 1.
Dang thiic xay ra chang han khi x = 2006. Vay gia tri nho nhit eiia fix)
la I.
4.15. a) Vdi X > 0 thi hidn nhien x + Ixl > 0. Vdi X < 0 thi X + IxU X - X = 0. b)x +
1 ^/7^X + 1 = X + ^-2) n^l^-2
1
+ ^ " 2 >0.
vay v x + Vx2 - x + 1 xac dinh vdi mpi x. 4.16. Ban An giai nhu vay la sai. Sai lim ciia ban An la khdng de y didu kien ciia cac sd a, b trong bit ding thirc gifia trung binh edng va trung binh nhan ^ i ^ > Va6 la a > 0, b>0. Trong bai nay x va 1 - x chi khOng am khi x G [0 ; 1], Ldi giai dung la ; 1 x(l - x) < - o
.2 , _ 1 ^ 2 I . . { 1 -x^ + x < - o x"" - x + - > 0 o X - -
ding thiic nay hien nhien diing vdi mpi x. 4.17. a) Vdi a > 0, 6 > 0 ta cd a + b> 24ab >Q',ab + \ > 2'Jab > 0. Tir dd suy ra ia + b)iab + 1) > 24ab.24ab
= 4ab.
Ding thiic xay ra khi va ehi khi a = b = 1. b) Vdi fl > 0, 6 > 0, c > 0, ta cd a + b + c> 3lfabc > 0 ;
ab + bc + ca> 3 > / f l V ^ > 0.
TOf dd suy ra
ia + b + c)iab + be + ca) >3l[abc .3^4^^c'^ Dang thiic xay ra khi va ehi khi a = b = c.
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= 9abc.
> 0 , bit
4.18. Vdi fl > 0, 6 > 0, c > 0 thi
1+ ^>2J£>0; b
\b
Tir dd suy ra | 1 + —
l+:^>2/>0; c
1+^>2J^>0. a
\c
c ii
a )
\a
2 la b c \ b c a
4.19. Do 0 < a < 6 nen -r < 1 suy ra b 1 P a\- + -r = 1 + - < 2 tiic la « < a b; b
(1) 1 1
-a^l T • . 1 1 ^ 1 L a i CO — + — > 2- —- n e n
a
b
\ab
2
< 4ab.
(2)
1 1
Do 0 < a < 6 nen 4ab < —-— < b.
(3)
Tit (1), (2),va (3) suy ra didu cin chiing minh. 4.20. a) x2 + - ^ > 2 / x 2 . ~ = 8 . Ding thiie xay ra khi x = ± 2. X'
vay gia tri nho nhit ciia fix)
la 8 khi x = ± 2.
b ) D o O < x < l n e n l - x > 0 . Tacd i = i ^ + 1 ; - ^ = - ? ^ + 2; X X 1-x 1-x 1-x 2x i 2 + -1 - x +, 3 > 2 VJ iX- ^ . 1- -^x + 3 = 2V2 + 3. X 1-x . Ding thiie xay ra khi
1-x
2x 0 < X < 1 tde la X = - 1 + V2. 1 - x va vay gia tri nhd nhit ciia g(x) la 2V2 + 3 khi x = - 1 + V2. 4.21. Do 0 < X < - nen fl - 2x > 0. Ta ed xia - 2x)2 = i . 4 x . ( a - 2x)(fl - 2x) < i . '2a v3
4x + a - 2x + a - 2x
2a^ 27
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127
Dang thiic xay ra khi va chi khi 4x = a-2x, tiic la x = —. 6 . 2a vay gia tri nhd nhit ciia y la ^ khi vachikhix= —. 2/
6
4.22. Gpi canh hinh vudng dupe eit la x (0 < x < l5, don vi: xentimdt) Thd tich V ciia cai hdp la V = x(80 - 2x)(50 - 2x). Khi dd ta ed
50 cm
12V = 6x(80 - 2x)(100 - 4x) ,3
< I 6x + 80 - 2;^ + 100 - 4x | ^ ^^3 80 cm-
Suy ra V < ^
hay V < 18 000.
Hinh 4.1
Dang thiie xay ra khi va chi khi 6x = 80 - 2x = 100 - 4x tiic la x = 10. Gia tri Idn nhit cua V la 18000 cm^ khi x = lO(cm). vay phai cit di 6 bdn gde vudng ciia hinh chii nhat ban diu nhihig hinh vudng cd canh 10 em. Nhan xet. Ndu xdt 4V = 4x(80 - 2x)(50 - 2x) thi 4V la tich ciia ba thiira sd f\30 cd tdng khdng d6i (bing 130), ta vin cd bit ding thiic 4V < -— V 3
nhimg ding thiic khdng thd xay ra va khdng cd gii tri nio cua x thoa man 80 - 2x = 50 - 2x. 4.23. Hudng ddn. Ap dung bit ding thiie Bu-nhi-a-edp-xki. 4.24. Dat b + c = X, c + a =^ y ; a + b = z. Do a, by c duong nen x, y, z duong -X + y + z , X - y + z x + y-z a= ^ ; b= ~ ,c= ~ . Khi dd ta cd 2 2 2 -x+y+z x-y + z x+y-z \(x y x z y z ^ A= z^ + :^ + :r = - - +— +— +— +—+ 3 2x
2y
2z
>f(2.3-3) = | . Hpc sinh tu giai tie'p. 128
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4.25. (h. 4.2) Tacd 1
R
AB = IA+IB >24lAJB = 2y[of- = 2R AB = 2R<^ IA = IB = R. Liic dd tam giac OAB vudng can tai O, canh huydn AB = 2R. OA = OB= Suyra
R42.
S^^ > | . 2 / ? = i?2
Hinh 4.2
vay SQAB nho nhit bang f^ khi OA = OB = i?V2
Khi dd toa dd
A(;?V2 ;0) va S(0;/eV2). 4.26. a)Dung,vi22 + 2 + l > 0. b) Sai, vi (-3)^ - 3.(-3) - 1 < 0 nen - 3 la nghiem eiia bit phuong trinh da eho. c) Sai, vi ^2 + (1 + ^)a _ a + 2 = 2^2 + 2 > 0. 4.27. a) Khdng tuong duong, vi x = 2 la nghiem eiia bit phuong trinh thii nhit nhimg khdng thude tap xac dinh ciia bit phuong trinh thii hai. b) Tuong duong. c) Khdng tuong duong, vi x = 0 la nghiem ciia bit phuong trinh thd nhit nhung khdng la nghiem ciia bit phuong trinh thii hai. d) Tuong duong, vi khi x - 3 > 0 thi x2 > 0 nen x - 3 > 0 <^ x2(x - 3) > 0. e) Khdng tuong duomg vi x = -1 la nghiem eiia bit phuong trinh thii hai •' nhimg khdng la nghiem ciia bit phuong trinh thu" nhit. g) Tuong duong, vi x2 - 2x + 2 = (x - 1)2 + 1 > 0 vdi mpi x. 4.28. a) Didu kien : x = 2, tap nghiem 5 = {2}. b) Didu kien : x > - , tap nghiem S =
3 2:^00
c) Didu kien : x > 3, tap nghiem S = 0 . d) Didu kien :x^2,
tap nghiem S =
I-
u (2 ; + CO).
booktoan.com 9-BTOS10,NC-A
129
4.29. a) Ve trai ludn duong vdi mpi x > 2. b) Ve trai khdng am vdi mpi x. c) Gian udc ca hai ve cho x2 + (x - 3)2 din d^n 2 > 5. Didu nay vd li. d) Do Vl + 2(x + 1)2 > I va VlO - 6x + x'^ = yjl + ix-
3f
> 1-
4.30. a) Vd trai ludn duong vdi mpi x. b) Vd trii khdng am vdi mpi x. 2
c) Gian udc ca hai ve' cho x ludn duong. 4.31. a ) x ; i - l ;x^3. b) X > 1 ; x^2;x^3
•>
•>
Ve' trai cua bit d:mg thiic mdi nhan dupe
•,x^4.
4.32. Sai lim ciia ban Nam la khdng dd y den didu kien xac dinh ciia phuong trinh D = [2 ; +co). Hai ve' ciia (1) ehi khdng am khi x e D chii khdng phai vdi mpi X e IR. Vi vay, khi tim ra x < 1 cin phai dd'i chi^u vdi didu kien X G [2 ; +oo) dd k^t luan bit phuong trinh (1) vd nghiem. 4.33. Sai lim ciia ban Minh la nghl ring ~ < — <^ b i) , {ab<^ >0 <^ \ ' ' hoac , a b ^b [a > b ' [a
(1)<^
x + 5>0
h)S =
f -co
\ 130
X <--
7
7 3 <=> - 5 < X < - - • X > ~5
///////////////mm
4.34. a) 5 - I y ; + 00 .4
<=>
J
^/^l
4
v/iiii^/i^'/m^im/m/
4 _
booktoan.com 9-BTDSiaNC-B
c)S
=
\-c^;-
y//////////////////////////////////' d) [5 ; +00).
//////////////////////////A
4.35. a) 5 = [-3 ; -2]. Bit phuong trinh da cho tuong duong vdi he x>-3 x + 3>0 X + 4 > 0 tiic la X > ^ hay - 3 < X < - 2 . X < -2 x + 2 <0 b)5 = ( - c o ; - 4 ) u ( - 3 ; - 2 ) . c) V ( x - l ) ^ ( x - 2 ) >0. :)>/u^
(1)
• Ne'u X = I thi bit phuong trinh (1) dupe nghiem diing. • Ndu X ^ 1 thi (1) tuong duong vdi x - 2 > 0, tiic li x > 2. vay tap nghiem cua (1) la 5 = {1} L; [2 ; +oo). d) V 2 x - 8 - V 4 x - 2 1 > 0 o V 2 x - 8 > V 4 x - 2 1 . 21 13 Didu kien : x > — , khi dd ta ed 2 x - 8 > 4 x - 2 1 , tiic iax < —. 21 13 Ke't hop vdi didu kien tren din de'n — < x < —. Vay tap nghiem '21 13' s= 4 ' 2 2x < -
3x + - < X + 2
4.36. a)
6x-3
<=> <
<=>
< 2x + 1
X< 1+
^<2
Bidu didn tap nghiem -^ "^ I ~*^' TTT J ^""^^ ^^^^ ^^ (phin khdng bi gach) 2_ 10
m\\\\\\\\\\\\wwmmM'
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131
b)5 = | y ; 1 3 23 2
mmmmmmmmmm
Wlllllllllllh
4.37. a) Ta cd mx > m
(1)
Neu m > 0 thi (1) <^ X > m ; tap nghiem S~{m;
+co),
N^u m = 0 thi (1) o 0.x > 0 ; tap nghiem 5 = IR. Neu m < 0 thi (1) <=> X < m ; tap nghifm
5 = (-co ; m]. (2)
b) Bie'n ddi vd dang (m - l)x > m + 2. Ndu m > 1 thi (2) <=> x > ^ "^ , . tap nghiem S = ——- ; +co m-1 •^'=i^m-1 Ndu m = 1 thi (2) o 0.x > 3, tap nghiem 5 = 0 ; N^u m < 1 thi (2) o X <
m+2 . tap nghiem S = m-1
—00 :
m+2
'•m-1
e) Bien d6i vd dang 1
1
1 "^
. X < iab + be + ca) a+b + -b+c + c+a a+b + b+c + c+a N^u
a+b
+
b+c
+
c+a
> 0 thi tap nghiem S = (-oo •,ab + bc + ca].
N^u
1 1 = 0 thi tap nghiem 5 = R. + + a+b b+c c+a
Ne'u
1 + .+ < 0 thi tap nghiem S=[ab +be + ca; +oo). a+b b+c c+a
1
d) Bie'n d6i vd dang xia + b) 0 thi 5 = Ne'ua +
132
ft
-00 ;
b.
a-b^ a +b
i ^ — ^ ; + o o ). a +b
booktoan.com
Ne'u a + b = Ovaa>bth\S
= R.
Ne'u a + b = Ovka 0
[x + 1 > 0.
(tha'y ngay x = -1 la nghiem cua (I) nhung khdng la nghiem ciia (II)). Suy luan diing la \AB>0 \B>0 < <=>A = Ohoae [A>0 ' [A>0 4.39. Ta cd | x + 4m2 < 2mx + 1
|(1 - 2m)x < 1 - 4m^
(1)
[3x + 2 > 2x - 1
|x > -3.
(2)
Ne'u m<— thi(l)<=>x -2. Ke't hpp vdi didu kien m < —, ta cd -2 < m < —• Ne'u m = — thi (1) cd dang 0.x < 0 (luon diing vdi mpi x € M), nen he (I) ludn cd nghiem x > - 3 . Ne'u m > — thi (1) <=> X > 1 + 2m, nen he (I) ludn ed nghiem X > I + 2m. Vay khi m > -2 thi he (I) ludn ed nghiem. 4.40. He vd nghiem khi -2 < m < 3. 4.41. a) Hpe sinh tu lap bang xet da'u (3x - l)(x + 2) > 0 khi x < -2 hoac x > - ; (3x- l)(x + 2 ) < 0 k h i - 2 < x < -• 2-3x b) jX — 1
^,^. 1 2 2-3x ^,.. 1 . 2 > 0 khi - < X < ~booktoan.com ; < 0 khi X < - hoac x > —• - •-'
^
'^X
I
D
^
133
c) Lap bang sau : -00
X
-
2
-x+1
+
x+2
-
3x+l
-
(-x+l)(x + 2)(3x+l)
+
0
0
1 3
1
+00
+
+
+
+
+ +
-
0
+
-
0
+
0
-
0
-
vay (-X + l)(x + 2)(3x + 1) < 0 khi - 2 < X < - - hoac x > 1 ; (-X+ l)(x + 2)(3x+ l ) > 0 k h i x < - 2 h o a c - - < x < 1. ,, ^ , 2 + x • 5x - 6 , ^ . , d) Ta CO 2 _ = _ . Lap bang sau :
X
-00
2
6
-
-
-
5x-6
-
3x-2
-
0
+
5x-6 3x-2
+
II
-
0
+CO
+ +
0
+
,,^ ^ 2+x ^,^.2 6 ^ 2+x ^,^. 2^ 6 vay 2 - O k h i x < - hoacx>T. 3x-2 3 5 3x-2 3 5 4.42. a) 9x2 - 1 = ^^^ ^ ^-^^^^ _ j ^ ^^^ ^.^^^ ^^ j^p ^^^^ ^^^ ^^^ ^^ ^^^^ ^^^^ 9x - I < 0 khi - ~ < X < - ; 9x2 - 1 > 0 khi X < - - ^^^^ x > - . b) -x'' + 7x - 6 = - (x - l)(x - 2)(x + 3). Hpc sinh tu lap bang xet diu va nhan dupe -x^ + 7x - 6 < 0 khi - 3 < X < 1 hoac x > 2 ;
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_;^•3 + 7^ _ 6 > 0 khi X < - 3 hoac 1 < x < 2. 134
c) x^ + x2 - 5x + 3 - (x - \)\x + 3). x^ + x2 - 5x + 3 < 0 khi X < - 3 ; x^ + x^ - 5x + 3 > 0 khi X > - 3 va X ?t 1. d) x2 - X - 2V2 =
I-V1 + 8V2
X -
1 + Vl + 8V2
2 o /^ A UK- ^ - Vl + 8v^ 1 + V1 + 8V2 X - X - 2V2 < 0 khi -— 0 khi X < ^hhoac x > ^^r 2x
4.43. a) Bi^n ddi bidu thiic vd dang
T Hoc sinh tu lap bang xet (3-x)(3 + x) • . .1^ & da'u. Ke't qua dupe bidu thiic duong khi x < - 3 hoac 0 < x < 3 ; bieu thiic am khi - 3 < X < 0 hoac x > 3. x2 - 6x + 8 _ (x - 2)(x - 4) . Lap bang xet diu sau : ^ 2 ^ 8 ^ , 9 - ( x - l ) ( x + 9)
b)
X
-00
-9
I
2
-
-
-
x-4
-
-
-
-
x-1
-
-
+
+
+
x+9
-
0
+
+
+
+
(x-2)(x-4) (X - l)(x + 9)
+
II
-
vay
II
+
0
+
+
x-2
0
0
4
-
0
0
+ 0
0
+
+
x^ - 6x + 8 0 khi X e (-00 ; - 9 ) w (1 ; 2) w (4 ; +00). x2 + 8x - 9
c) Bie'n doi bieu thiie vd dang ix + 2f
x'ix'^2)
TU dd, bidu Ihiic da cho se
duong khi x G (-QO ; -2) u (-2 ; -V2 )
(-V2 ; 0) u (0 ; V2).
booktoan.com 135
X+ 1 - 1 X + X+ 1 d) Ta cd -^r ' =< -x-2 X + X+ 1
khi X > - 1
khi X < - 1 .
.X + X + 1
Diu eua bidu thde tren hoan toin phu thupc vao diu cua tii thiie (vi x^ + X + 1 > 0 vdi mpi x). Vi vay : Ix + ll - 1
> 0 khi X G (-00 ; - 2 ) w (0 ; +co).
x2 + X + 1
4.44. a) Tap nghiem 5 = (-oo ; -1) u
V2;||.
b) Bie'n ddi bit phuong trinh vd dang 4 5'
Tap nghiem S =
5x + 4 r < 0. 3x + l
_\ 3J"
4.45. a) Dua vao tmh chit lai + \b\ = \a-b\
<^ab<0,
va dd y ring (5 + x) - (x - 3) = 8 ta ed 15 + xl + Ix - 31 = 8
o (5 + x)(x - 3) < 0 O -5 < X < 3.
Cha y. Hpe sinh ed thd giai bing each chia thanh cac khoang dd pha diu gia tri tuyet ddi nhung ldi giai se dai hon. b) Dua vao tinh chit lai = a o a > 0, ta cd X - 5x + 6 = x 2 - 5 x + 6 < = > x 2 - 5 x + 6 > 0 < = > x < 2 hoac x > 3. 2x - 1 khi X > e) Ta ed |2x - ll = 1 - 2x khi X < - -
booktoan.com 136
Ne'u X > - thi |2x - l| = X + 2 o
2x - 1 = x + 2 <^ x ^ 3 (thoa man
didu kien x > - ) . Ne'u X < - thi |2x - ll = X + 2 o
1 - 2x = X + 2 <» X = - - (thoaman
didu kien x < —). vay tap nghiem cua phuong trinh la 5 = -— ; 3 d) Tap nghiem 5 = {-3 ; 2 } . 4.46. a) |3x - 5| < 2 <=> - 2 < 3x - 5 < 2 o b)
1 < X < -^.
2 - x ^ , 2-x 2-x > 2 hoac < -2. > 2 <» x+1 x +1 x +1
Trudng hop ^ — ^ > 2 o x+1 T,v
.
Truong hpp
2 - x ^ ^
- ^ > 0 <=> - 1 < x < 0. x+1 4 + x
r < - 2 <=> < 0 <:^ - 4 < X < - l . x+1~ x+1
vay tap nghiem S = (-4 ; -1) u (-1 ; 0]. e) Phan chia hai trudng hpp x ^ 2 va x < 2 . Tap nghiem 5 = I -oo ; - ). d) Ta cd X
+ 11 =
x + 1 khi X > - 1
X khi X > 0 X —
~x - 1 khi X < - 1 ;
-X khi X < 0.
Gpi bit phuong trinh da eho la (1). • Ndu X < - I thi (1)<^ - x - 1 < - x - x + 2 <^ x < 3. Ke't hpp vdi didu kien x < - 1 , ta dupe x < - 1 . • Ne'u - 1 < X < 0 thi ( l ) o x + l < - x - x + 2<=>x
• Ne'u X > 0 thi (l)<=>x + l < x - x + 2 < ^ x < l . Ke't hpp vdi didu kien x > 0 , ta dupe 0 < x < 1. vay tap nghiem eua (1) la 5 = (-oo ; 1]. 4.47. a) 2(x + y + 1) > X + 2 o X + 2y > 0. Midn nghiem cua bit phuong trinh la niia mat phang (phin khdng bi gach, khong kd bd) trong hinh 4.3.
^^:N^i^
Hinh 4.3
Hinh 4.4
b) 2(y + X) < 3(x + 1) + 1 <::> X ~ 2y + 4 > 0. Midn nghiem ciia bit phuong trinh la niia mat phang (phin khdng bi gach kd ca bd) trong hinh 4.4. c) Midn nghiem la nua mat phing (phin khdng bi gach, khdng kd bd) trong hinh 4.5. >
y y^5
II K 1
m /nn
7////7/////C
^'' •:
0
3
X
ii. ijiinl/
Hinh 4.5
Hinh 4.6
d) Midn nghiem la nua mat phing (phan khong bi gach ke ea bd) trong hinh 4.6-
booktoan.com 138
4.48. a) Midn nghiem la phin khdng bi gach (khdng kd bien) trong hinh 4.7. y A 3
^- = 3
0
Hinh 47
A-
Hinh 4.8
b) Midn nghipm la phin khdng bi gach trong hinh 4.8 (khdng kd tia At) 4.49. a) Midn nghiem la phin khdng bi gach (khdng kd bien) trong hinh 4.9. b) Midn nghiem la midn tam giac ABC (khong kd hai canh AB, BC) trong hinh 4.10. ..
Hinh 4.10
Hinh 4.9 X-1 <1
r_l<;C_l
fO
4.50. Ta cd ^ ^ \ ^ , ^ ^ . [|y + l| < 2 [-2 < y + 1 < 2 [-3 < y < 1. Midn nghiem la midn hinh chu nhat DABC (khdng kd hai canh AB va CD) dhinh 4.11.
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139
fJ.JJJ-
[////
/ / / / / / / / / / D '; f
/ . / / / / / / / /
n~rtt/-f-rttT7ii}H\ y=i
0
c
mmmm Hinh 4.1}
4.51. a) Xem hinh 4.12, midn nghiem la hinh ngu giac ABCDE.
Hinh 4.12
b) Bieu thire 7 -= 2x - 2y + 3 dat gia tri nho nha't tai mpt trong cae dinh eua ngu giac ABCDE. Dung phep thii true tiep, ta thay 7" - 2x - 2y + 3 dat gia tri nho nhat bing -17 tai x - 0, y - 10 (tai didm D).
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140
4.52. a) Sd gid lam viec trong mdi ngay eua M^ la 3x + y. Sd gid lam viec trong mdi ngay cua M2 la x + y. Theo bai ra ta cd he bit phuong trinh '3x + y < 6 x+y<4 (I)
x>0 y >0.
Midn nghiem (5) eiia he (I) la mien tii giac OABC (h.4.13). b) Sd tidn lai ciia xi nghiep mdi ngay la 7 = 2x + l,6y (trieu ddng)
Hinh 4.13
c) T = 2x + l,6y dat gia tri Idn nhit tai mpt trong cac dinh eiia tii giac OABC. Dung phep thii true tie'p, ta tha'y 7 = 2x + l,6y dat gia tri Idn nhit k h i x = 1 ;y = 3 (didm B).
*
vay dd sd tidn lai Idn nhit (6,8 trieu ddng), xi nghiep cin san xuit mdi ngay 1 tin san phim I va 3 tin san phim II. 4.53. a) Tam thiic da cho ed a = 2 > 0 va biet thiic A' = 1 - 10 = - 9 < 0, nen tam thiic ludn duong. b) Tam thiic da cho ed a = - 1 va biet thiie A = 1 > 0, va cd hai nghiem Xi = 2, X2 = 3. Suy ra tam thiic duong trong khoang (2 ; 3) va am trong eac khoang (-00 ; 2) va (3 ; +co). c) Tam thu'c da cho c6a-2, X ^
biet thiic A = 0 nen tam thiic duong vdi mpi
r-.
booktoan.com 141
•3
d) Tam thiie da eho cd a = -4, biet thiie A' = 8 > 0 va ed hai nghiem J 1 +V2 V2-1 ,u ' X[ = — , X2 = — - — . nen tam thiic duong trong khoang I + V2 V 2 - O V
^V2-l V
'
2
;+co
.
2 2
^ , . ' ,U ' ^ va am trong cae khoang -00
1+ ^ 1 ;
X 1 va
e) Tam thdc da cho cd a = >^ va biet thiie A = (V3 + 1)^ - 4^/3 = (V3 - 1)^ > 0, tam thiie ed hai nghiem Xi = - 1 , X2 = —!=• Suy ra tam thiie duong trong eac khoang (-00; -1),
I
;+00
va am trong khoang
'V3J
Chii y : Nhan x6ta-b
+ c = 0 nen tam thiic co hai nghiem -
1
_
^ _
1
Tii dd ap dung dinh li vd diu tam thiic. f) Tam thiic c d a = l v a ( 3 + Z? + c = 0 , nen tam thiie ed hai nghiem X] - - V s , X2 = 1. Suy ra tam thiic ludn duong trong cac khoang (-00; -V5), (1 ; +co) va am trong khoang (-V5 ; 1). g) Tam thiic da cho co a = -0,3 < 0, biet thire A = -0,8 < 0, nen tam thitc ludn am vdi mpi x. h) Tam thiic da cho co a = \, A-(V7-l)2-4>^-8-2V7-4V3 = 2 ( 2 - V 7 ) + 4(l - V 3 ) < 0 . Nen tam thiic ludn duong vdi mpi x. x-7
_
. ,
. 2
4.54. a) Dat Aix) = 4x2 _ j g ^ _^ j 2 . Tam thiic 4x2 _ j ^ ^ _^ ^2 cd hai nghiem =^.x.X2 = 4. 4'
142
booktoan.com
Lap bang xet diu Aix) : 3
x
-00
4
-
x-1 4x2 __ j ^ ^ ^ j 2
Aix)
-
+
0
-
4
7
0
-
+CO
+
-
0
+
+
.
I
-
0 .
(3
\
Tit bang xet dau ta thu duoe Aix) > 0 trong eac khoang — ; 4 va (7 ; +oo) \4 / 3 va Aix) < 0 trong cae khoang (-oo ; — ) va (4 ; 7). l^^'^^—.Tam thiic -x2 + 5x - 7 ed a = - 1 < 0 vi biet -x2 + 5x - 7 thiic A = -3 < 0 nen tam thiie ludn luOn am vdi mpi x. Suy ra JB(X) > 0 3 o 1 Ix + 3< 0 <=> X < - — va b) Dat fi(x) =
Bix) < 0 o llx + 3>0<=>x>-—c) Dat C(X) =
3x-2 x^ - 3 x 2 + 2
3x-2 (X - l)(x2 - 2x - 2)
Lap bang xet diu (HS tu lap), ta thu dupe : C(x) > 0 trong cac khoang (-oo ; 1 -V3 ), ( - ; 1) va (1 + V3 ; +co). C(x) < 0 trong cac khoang 1 - V3 ; d) Dat Dix) =
va (1 ; 1 +V3).
x^ + 4 x - 1 2 V6x2 + 3x + V2
Ta thiy tam thire V6x2 + 3x + V2 > 0 vdi mpi x, nen diu eiia Dix) cung 2
diu vdi diu ciia tam thiie x + 4x - 12. Suy ra 0(x) > 0 trong cae khoang (_oo ; - 6 ) va (2 ; +oc^), Dix) < 0 trong khoang (-6 ; 2).
booktoan.com
143
e) Dat £(x) =
x2 - 3x - 2 -X^ + X - 1
Ta tha'y -x2 + jc - 1 < 0 vdi mpi x , nen £(x)
trai dau vdi dau tam thiic x - 3x - 2. Suy ra : £(x) > 0 trong khoang
'3-4vi
£(x) < 0 trong cac khoang
— 00
3 + Vl7 ;
3-Vl7'l . r3 +Vl7 z
va
r
;+co
.
Ar,-. F. ^ ^^ - 5x + 4 (x - l)(x2 + x - 4) f) Dat F(x) = — z = ^-^ x^ - 4x^ + 8x - 5 (x - l)2(x2 - 2x - 5) Lap bang xet diu (HS tu lap) ta thu dupe : Fix) > 0 trong cac khoang
-i-Vn
;l~>/6
1;
-1 + Vl7
va(l + V6 ; +oo).
Fix) < 0 trong cac khoang ,(1-V6;1),
-00 :
-1 + Vl7
; 1 + N/6
.
4.55. a) Ta ed biet thiic A = (m + 1)2 - 4| m - ^ = m2 - 02m ^"7 + —. 2
7
-
4
Xet tam thiie/(m) = m -2m+ - , c6 a = \va biet thiic A' = - - < 0 nen fim) > 0 vdi mpi m. Vay phuong trinh ludn cd nghiem. C/?My .•TacdthdxetA = (m+ 1)^- 4f m - i ] = (m - 1)2 + 1 > 1 b) Ta cd A' = (m - 1)2 - (m - 3) = fm - | j
+ I > I > 0, nen phuong
trinh lu6n ludn cd nghiem. • Chii y : Ta ed thd sir dung dinh li vd diu tam thd:c bac hai dd lam bai tap nay, hpe sinh tu lam. c) Ta cd A = (m + 2)2 - 4\^-m + 2] ^ [^'^ 2J- '^ ~4~ ^ > 0, nen phuong trinh ludn cd nghiem. 144
booktoan.com
d)
*) Ne'u m = I phuong trinh cd nghiem x = - 1. *) Ne'u m ?^ 1 ta cd A = (3w - 2)2 - 4(m - 1)(3 - 2m) = 17/n2 - 32m + 16 = m^ + 16(m - 1)^ > 0,
nen phuong trinh ludn cd nghiem. Tom lai vdi mpi gia tri ciia m thi phuong trinh ludn ed nghiem. 4.56. a) Ta cd A' = 4^2 - 2(2m^ + 1) = -2 < 0, nen phuong trinh v6 nghiem vdi mpi gia tri eua m. 2
2
2
b) Ta cd A = (m + 1) - 2(m + m + \) = -m - 1 < 0, nen phuong trinh vd nghiem vdi mpi gia tri cua m. c) Ta cd A' = (m - 3) - (2m - 7m + 10) = - m + m - 1. Xet tam thiic/(m) = -m + m - \, c6 a = -\ va A = -3 nen/(m) < 0 vdi mpi m. Suy ra phuong trinh ludn vd nghiem. d) Ta ed A = iSm
- 1)2 - 4(m2 - V3m + 2) = -m^ + 2Sm
- 1
= -(m - V3 ) - 4 < 0 nen phuong trinh v6 nghiem vdi mpi gii tri eiia m. 4.57. a) Ta ed A' = 4 - (m - 5) = 9 - m va tam thiic cd a = 1 > 0. Tam thiic ludn duong khi va chi khi A' = 9 - m < 0 o w > 9. b) Tam thiie da eho cd biet thiic A = (m + 2)^ - 4(8m + 1) = m^ - 28m - mim - 28) va c? = 1. Tam thiic ludn duong khi va chi khi A = mim - 28) < 0 <=> 0 < m < 28. c) Ta cd A' = 4 - (m - 2)2 = -m2 + 4m va he sd a = \. Tam thiic ludn duong khi va chi khi A = -m2 + 4m<0<=>m>4 hoac m < 0. d) *) Ne'u 3m + 1 = 0 thi m = - ~ . Khi dd bidu thiic ludn duong vdi mpi x. *) Ne'u m^ -— thi tam thu'c da cho cd biet thiic A = (3m + 1) 2 - 4(/M + 4)(3m + 1) = (3m + l)(-m - 15) = -3nflO-BTDSlO.NC-A
46m -15 = -i3m^ + 46m + 15).
booktoan.com
145
Tam thiic ludn duong khi va ehi khi -1 a = 3m + \ > 0 m> <^ ^ 3 A<0 (3m + l)(m + 15) > 0
(*)
o m > - - hoac m < -15. Ke't hpp vdi (*) suy ra m > - - • Tom lai vdi m> — thi bidu thiie ludn duong vdi mpi x. 4.58. a) *) Khi m = 4 dd thiy bidu thiic khong luon luOn am vdi mpi x. *) Khi m^4,di tam thiic ludn am vdi mpi x, didu kien cin va dii la : fm - 4 < 0 (*)
A = (m + 1)2 - 4(m - 4)(2m - 1) < 0. 0
3
Ta cd A = -7m + 38m - 15, A < 0 khi va ehi khi m < - hoac m > 5. Kdt 3 hpp vdi i*), suy ra m < —. b) *) Khi m = -2, bieu thiic da cho trd thanh 5x - 4. Bidu thiic nay khdng thd ludn ludn am vdi mpi x. Vay m = -2 khdng thoa man. *) Khi m^-2
thi tam thiie ludn am khi va chi khi
fm + 2 < 0 < o [A = 25 + 16(m + 2) < 0
57 m<-—-. 16
c) Bieu thu'c ludn am khi va ehi khi m < - — . d) Bidu thiic ludn am khi va chi khi - - < m < - 1 . 4.59. a) xet tam thilcy(x) = 2x^ - 7x - 15 cd o = 2 > 0 va A = 49 + 120 = 169 = 13^ 3 nen tam thiic ed hai nghiem Xj = -—, X2 = 5. Do dd bit dang thiic cd tap _3 nghiem la : (-co; —] u [5; + oo) b) Nghiem bit phuong trinh la -— < x < — •
booktoan.com 146
10-8TDSiaNC-e
c) Tap nghiem bit phuong trinh la (- oo; 1 ] w [4 ; + oo). d) Bit phuong trinh dupe bie'n d6i thanh (2x + 3)^ > 0 nen tap nghiem la tap sd thuc R. e) Nghiem bit phuong trinh la 3 < x < 6. 4.60. a) Bit phuong trinh da eho tuong duong vdi bit phuong trinh : 2x-5 x^ - 6x - 7
1
^ n ^ (2x-5)(x--3y-(x^-6x-7) -. TTT ZT-. —. (X - 3)(x + l)(x - 7)
X - 3 < uo
. < U
x^ - 5x + 22 <0 (X - 3)(x + l)(x - 7)
(*)
Tam thiie x^ - 5x + 22 ed a = 1 > 0, A = -63 < 0, nen x^ - 5x + 22 > 0 vdi mpi X. Suy ra (*) tuong duong vdi (x - 3)(x + l)(x - 7) < 0. Lap bang xet diu : X
x-3
-
x+ 1
-
x-7
-
(x-3)(x+l)(x-7)
3
-1
.00
_ 3
0
0
7
+00
+
+
+
+
+
-
-
0
+
-
D
+
+
0
Tuf bang xet diu suy ra tap nghiem ciia bit phuong trinh da cho la : r = (-oo;-l)L^ (3;7). b) Bit phuong trinh dupe bie'n d6i tuong duong thanh : 11x2 + 5x + 6 x(x2 + 5x + 6) Suy ra tap nghiem la : 5 = (-co ; -3) w (-2 ; 0). e) Bit phuong trinh dupe bie'n ddi tuong duong vdi: (X + 1)(2 - X)
^Q
(X + l)(x2 - X + 1)
Suy ra tap nghiem la : 5 = (-co ; -1) u (-1 ; 2].
booktoan.com 147
d) Bit phuong trinh dupe bie'n ddi tuong duong vdi: x^ + X - 1 <0. ( x - l ) ( x + l)x Suy ra tap nghiem la : 5 =
-00 :
-i-V^
4.61. Bit phuong trinh dupe bie'n d6i thanh
w (- 1 ; 0) u
-1 + V ^ . /
2x + 9 < 0, vdi X ;t 0. ix - 2)(x + 2)
Suy ra tap nghiem cua bit phuong trinh da cho la 5=|^-oo;-^Ju(-2;0)u(0;2).
Do dd, gia tri nguyen khdng am cua x thoa man bit phuong trinh lax=l. 4.62. a) Nhan xet x = - l va x = 2 la nghiem cua phuong trinh x - x - 2 = 0. Ndu X ?^ -1 va X ?t 2 thi bit phuong trinh tuong duong vdi he [x^ - X - 2 > 0
[x < -1 hoac x > 2.
vay tap nghiem ciia bit phuong trinh da cho la 5 = [2 ; + oo) u {-1}. b)r = [-2;3I. 4.63. a) Phuong trinh x2 - 2x - 3 = 0 ed hai nghiem xj = -1 ; X2 = 3. Suy ra bit phuong trinh x^ - 2x - 3 > 0 cd nghiem la : Sj = (-oo ; -1) u (3 ; +oo). Phuong trinh x2 - 1 Ix + 28 = 0 cd hai nghiem Xj = 4 ; X2 = 7. Suy ra bit phuong trinh x2 - 1 Ix + 28 > 0 cd nghiem li : S2 = (-00 ; 4] u [7 ; +00). Nghiem ciia he bit phuong trinh la giao eua hai tap nghiem S^ vi 52, tiic la S = Si n S2 = (-00, -1) w (3 ; 4] u [7 ; +00). Bidu didn tren true sd :
-1
148
•Hi
§
booktoan.com
b) 1 < X < I . c) Ba't phuong trinh v6 nghiem ; d) 1 < X < 7. 4.64. a) Phuong trinh x - 4x - 5 = 0 cd hai nghiem Xj = - 1 ; X2 = 5, nen bit phuong trinh x2 - 4x - 5 < 0 ed tap nghiem la 5] = (-1 ; 5) Phuong trinh x2 - 6x + 8 = 0 cd hai nghiem x, = 2 ; X2 = 4, nen bit phuong trinh x2 - 6x + 8 > 0 cd tap nghiem la ^2 = (-co ; 2) w (4 ; +QO). Nghiem ciia bit phuong trinh 2x - 'i > 0 la 53 = | ; + co). Suy ra nghiem ciia he la giao ciia ba tap 5|, ^2 , 53, tiic la S = S^ r> ^2 n 53 =
| ; 2 J w(4;5).
Bidu didn tren true sd : iilliljlljlitimiiiiiiiiiii^^^^^^^ v/iiiiiiiiimi/iim/iim WWM'//'. m/lmmmmm milllllmlll'
/}Illlllllllllllm^ -1
b)S =
' 3
(^ 13"
T1
!lll"/"Wliiliil"ll'lll flilftllfmlfllfin -4
3 2
^3. 4
2
4
5
. Bieu didn tren true sd :
WliWIi M!!i!"llilli'll'lllllli^l"l"i'll'll'^li!!>l'il''!'""""i'', JllllllllllK Willillflllmlllllllllm^
3
5
16
13 2
4.65. Tap xac dinh cua ham sd y - fix) gdm cae gia tri x thoa man 3-3x
3 - 3 x - 15 + 2x + x'
-1 >0
-x2 - 2 X + 1 5 -x2 - 2 x + 1 5 ^ 0 ' x2 - X - 12
<=>
>0
-(x^ + 2 X - 1 5 ) -x2 - 2 x + 15 ^ 0 (X + 3)(x - 4)
<0
<=> ^ (X - 3)(x + 5) • » < x2 + 2 X - 1 5 x^booktoan.com + 2 X - 1 5 ^ 0. .x2 + 2 x - I 5 ; ^ 0
149
D^t Fix) =
ix + 3)(x - 4) ( x - 3 ) ( x + 5)
Lap bang xet diu P{x) : X
-5
—00
3
x-4
-
-
-
x-3
-
-
-
x+3
-
0
x +5
+ _
+
Pix)
3 3
A
3 0
0
+00
+
+
+
+
+
+
+
+
+
+
-
Tir bang xet diu suy ra tap xac dinh cua ham sd fix)
0
+
la :
(-5;-3] u(3;4]. 2
4.66. a) Phuong trinh x - 3x - 4 = 0 ed hai nghiem Xj = - 1 , X2 = 4, nen bit phuong trinh x - 3x - 4 < 0 ed tap nghiem la Sj = [ - 1 ; 4] Xet bit phuong trinh (m - 1 )x - 2 > 0 <:> (m - 1 )x > 2. *) Ne'u m - 1 = 0 thi bit phuong trinh tren vd nghiem. *) Ne'u m - 1 > 0 <:> m > 1 thi bit phuong trinh (1) ed tap nghiem la S2=[
m- 1
; +oo).
Di he cd nghiem, didu kien cin va du la 5, ri ^j ?=: 0 Hie la 2 1 3 ——j-<4<=> — < m - l < ^ m >—, thoa man didu kien m > 1 Vay m > — *) Neu m - 1 < 0 <=> m < 1 thi bat phuong trinh (1) cd tap nghiem la S^ =
—oo :
' m—1
booktoan.com 150
(1)
Dd he cd nghiem, didu kien cin va du la S,r^S.^0o >-lo-(m-l)>2<=i>m<-l, ^ m-1 thoa man dieu kien m < 1. Vay m < - 1 . Tom lai cac gia tri cua m dd he cd nghiem la m e(-oo ; -I] KJ
3 - + C 0
b) Tap hpp cac gia tri m thoa man bai toan la :
4.67. a) -2 - V2 < m < -2 + V2 . b) Ne'u m = 1, phuong trinh cd nghiem x = 8 Ne'u m^ 1, dd phuong trinh cd nghiem didu kien cin va du li : A' - (m + 3)2 -im-
1)(2 - m) > 0 <^ 2m2 + 3m + 11 > 0.
Ta thiy tam thiic/(m) = 2m2 + 3 m + l l c d a = 2 > 0 v a A = -79 < 0 nen fim) > 0 vdi mpi m. vay phuong trinh ludn ludn cd nghiem vdi mpi gia tri eiia m. 4.68. a) m> 1. b) Khdng tdn tai m di bit phuong trinh nghiem dung vdi moi x. e) Ta thiy tam thiic x2.- 8x + 20 ed a - 1 > 0, A' - 16 - 20 = -4 < 0. 2
Suy fa X - 8x + 20 > 0 vdi mpi x. Do do bai toan tro thanh tim cac gia to m dd bit phuong trinh mx + 2(m + l)x + 9m + 4 < 0 (*) diing vdi mpi x. Ne'u m = 0 bit phuong trinh (*) trd thanh 2x + 4 < 0, bit phuong trinh ehi nghiem diing vdi x < - 2, nen m = 0 khong thoa man. Ndu m ?^ 0. Dd bit phuong trinh (*) dung vdi mpi x thi didu kien cin va dii la ;
Jm <0 IA'
=im + 1)2 - m(9m + 4) < 0.
booktoan.com 151
2 1 1 Ta thay tam thiic A' = - 8m - 2m + 1 cd hai nghiem Vam^ ='^, tn2= j nen A' < 0 <=> m < - - hoac m > - Ke't hop vdi didu kien m < 0. suy ra 2 • 4 '^ 1 cac gia tri cin tim ciia m la m < d) m > 5. 4.69. a) Phuong trinh cd hai nghiem am phan biet khi va chi k h i : A' =im + 1)2 - (9m - 5) > 0 2=-ini
m - 7m + 6 > 0 <=>
+ l)<0
Jm > - 1 5 m>-
ac = 9m - 5 > 0 m > -1 <=>
5 Im > —
<=> m > 6 hoac — < m < 1.
m > 6 hoac m < 1 vay cac gia tri cin tim cua m la m
-;1|
w(6;+co).
b)m e (-oo;-3) u ( 2 ; 6 ) . 4.70. a) + Vdi m - 2, phuong trinh da eho trd thanh :
V2
2
Phuong trinh ed hai nghiem, nen khdng thoa man yeu eiu eiia diu bii. + Vdi m ?^ 2, dat / = x2 > 0, ta dupe phuong trinh fit) = im- 2)^2 - 2(m + l)r + 2m - 1 = 0.. (*) De phuong trinh da eho cd dung mdt nghiem thi phuong trinh (*) hoac cd nghiem kep t = 0 hoac cd mpt nghiem am, con nghiem thd hai bing 0. Bay gid xet t = 0. Khi dd /(O) = 2 m - l - 0 o m - - .
Thay
m=~
vao (*) ta dupe : fit) = t\--t-3\
= ^ ^
"r-0 t = -2.
vay m = - la gia tri cin tim (dd phuong trinh da cho ed mpt nghiem).
booktoan.com 152
,^
7 + 3V5
b) m =
(\
- . Hudng ddn. Rd rang vdi m = 2 phuong
. m e - ;2
trinh cd hai nghiem x = ± - ^ .
V2 Vdfi m ?^ 2.
Di phuong trinh da eho cd diing hai nghiem thi phuong trinh (*) hoac ed nghiem kep duong hoac ed mdt nghiem am va mpt nghiem duong. - Phuong trinh (*) cd hai nghiem trai diu khi va chi khi ac < 0 tiic la im ~ 2)(2m - 1) < 0 hay - < m < 2 - Phuong trinh (=*') ed nghiem kep duong khi va chi khi A' = 0 va
--->0. 2(3
0 1 + 3J5 A' = -m2 + 7m - 1 = 0 <^ m ~ ; b m+1 > 0 <=> m < - I hoac m > 2. 2a m- 2 Chi cd m = c) 2 < m <
r
thoa man hai didu kien tren. Hudng ddn. Tim m de phuong trinh fit) = 0 ed hai
nghiem dUOng phan biet. Dieu kien cin va dii la A' > 0 , 5 > 0 va P > 0. 4.71. a) Phuong trinh dupe bidn ddi thanh 3(3x - 2) + V3x - 2 - 4 - 0.
(*)
Dat t = V3x - 2 > 0, khi dd (*) trd thanh 3/2 + ^ - 4 ^ 0 giai ra ed hai 4 nghiem t^ = 1, ^2 = - : T Do r > 0, nen ehi liy ; - 1. Vay (*) <:> V3x - 2 = 1 o x = 1. Phuong trinh da eho cd nghiem diiy nha't x = 1. b) X = 3. Hudng ddn. Phuong trinh tuong duong vdi he : -x^ + 2x + 4 = (x - 2)'
[x - 2 > 0 booktoan.com 153
c) X =
. Hudng ddn. Phuong trinh da cho tuong duong vdi he |x2 - 2x - 3 - (2x + 3)2 |2x + 3 > 0
d) X = - 3 . Hudng ddn. Phuong trinh tuong duong vdi V(9 - 5x)(3 - x) = 9 - X
'<-\ 4.72. a) x = - l , x = 4 . b) X G {1 ; 3}. Hudng ddn. Dat x2 - 4x + 10 = r, r ^ 0. c) X G j J ; 2[. Hudng ddn. Nhan xet x = 0 khdng la nghiem ciia phuong trinh, nen ta chia ca tii va miu ciia ve' trai ciia phuong trinh cho x ta dupe phuong trinh tuong duong : 2 2x +
13 5
= 6.
2x + - + l
X
X
2
13
+ = 6, Phuong trinh nay ed dang • y-5 y+1 trong dd 2x + ~ = y Tir dd giai dupe y = 1 va y = 5,5 d) X = —(1 - V2 ± V2V2 - 1). Hudng ddn. Cong vao hai vd ciia phuong trinh bidu thiic 2x.
x-1 2
Tii dd di de'n : Dat t =
• X
-\
X-1
N2
-2
x-1
= 1. •}
dupe phuong trinh f - 2r - 1 ^ 0 . 1.
[33^
fSJ
-
4.73. a) Xj = J — - X2 = - V ^ • Huang ddn. Phuong trinh dupe bie'n ddi thanh 2x" + 3 - 5 V^/ 2x^ + 3 - 6 - 0 .
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154
(*)
Dat t = ^2x2 + 3 > 0. Khi dd (*) trd thanh r^ - 5/ - 6 = 0 va ed hai nghiem fj = - 1 , ^2 = 6. Do r > 0, nen chi liy t = 6. b) X = 3 ; X = - - . Hudng ddn. Dat t=^2x^
+ 3x + 9 .
c) X = 0 ; X = 2. Hudng ddn. Dat r = V81 - 7x^ d) X = 1 ; X = - . Hudng ddn. Dat t = ^2x^ - 3x + 2 . 4.74. a) X =
yfvj -3
h)x=
-2-yl2;
c)xG 1-1,-3}; d)x =
-15±Vi65
4.75. a) Phuong trinh tuong duong vdi: (I)
Giai he
x^ - (2x - 1) = 0
hoac (II)
x2 + (2x - 1) = 0 2x - 1 < 0.
2x - 1 > 0
x =l 1 o x = 1. (I): ' ^ 2 Xi=-l-y/2;
Giai he (II) :
X2 = - 1 + V 2
1 ' ^ 2 <=> X = - 1 - V2 hoac X = - 1 + V2.
vay phuong trinh ed cac nghiem : x = 1 ; x = - 1 ± V2 . b) X = 1. Hudng ddn. Phuong trinh tuong duong vdi: 'r2 X- - 2x - 3 = x2 - 2x + 5 x2 - 2x - 3 > 0
hoac
-(x2 - 2x - 3) = x2 - 2x + 5 x^ - 2x - 3 < 0.
booktoan.com 155
4
c) X = — ; X = 2. Hudng ddn. Phuong trinh tuong duong vdi: (2x-3)2=.(x-l)2 d ) x = l ± V 6 , x = l ± V ^ . Hudng ddn. Phuong trtnh tuong duong vdi: x2 - 2x - 3 = 2 hoac x2 - 2x - 3 = -2. 4.76. a) 5 < X < 10. Hudng ddn. Dua phuong trinh vd dang :
lv:rn"-2|+|vr^-3|=:i.
7 b) - < X < 7. Hudng ddn. Phuong trinh dupe dua ve dang |Vl4x - 49 + 7| + |Vl4x - 49 - 7| = 14.
0 Ixl = - . d) X
V2 1 1 —-;—(V6-V2)r Hudng ddn. Ne'u x nghiem dung phuong
trinh thi —]= < x < -?= nen V2 V2
A/TT?
> |;t|, nghia la x + yjl-x^
> 0.
vay ta ed the gia thiet x < -T= va phuong trinh trd thanh : V2 x + V l - x 2 = V2(l-2x2). Mat khae 1-2x2 ^ ^^j ^ x^ + x)(Vl - x2 - x), nen ta. cd thd dua phuong trinh da eho vd : (X + V l - x2 ) V l - x 2
- ; c - - L U o .
4.77. a) -6 < X < -4 + V2 . Hudng ddn. Bit phuong trinh tuong duong vdi he : |-x2 - 8x - 12 > 0 x + 4 <0 b) XG
1^ '•^n
. |-x2 - 8x - 12 > (x + 4)2 x + 4>0.
w (4 ; + oo). Hudng ddn. Bit phuong trinh tuong duong vdi: 4x + 2 > 0 5x2 +61^->o 5x2 + 6 1 J C < ( 4 X + 2 ) 2 .
booktoan.com 156
c) X G (-00 ; 0) L; [1 ; 2]. Hudng ddn. Bit phuong trinh tuong duong vdi: fx^iO
lx(V2-x+2x-3)>0. d) X G
-I-
u
1 ; — I. Hudng ddn. Bit phuong trinh tuong duong vdi:
3x2 - 3 > 0
(2x + 3)[3(2x - 3) ^ V3x2 - 3] < 0. 4.78. a) Bit phuong trtnh tuong duong vdi he : x + 3>0
x + 3>0
l-x>0
o
^x < 1 2
x^ - 3x - 2 > 0. ^
X + 3 < (1 - x)^
Ttr dd suy ra tap nghiem bit phuong trinh la 5 = b) 3 < X < 5. Hudng ddn. Bit phuong trinh da cho tuong duong vdi he : [-x2 + 6x - 5 > (8 - 2x)2 ^^^^ | - x 2 + 6x - 5 > 0 •
8-2x>0
l8-2x<0.
1 "l c) 5 = 0 ; — W (4 ; +oo). Hudng ddn. Bat phuong trinh tuong duong
vdi he : (4x + 2)2 >5x2 +61x 5x2 + 6 1 x > 0 4x + 2 > 0. d) 5 = R. Hudng ddn. Bit phuong trinh da cho tuong duong vdi: ,x-x>x-2
x 2 - X > X - 2 <=> { _ x^ - X > 0
|x-x^>x-2 hoac
x^ - X < 0.
booktoan.com 157
4.79. a) * Ndu -5 < x < 0 bit phuomg trinh ludn ludn diing. * Xet X > 0. Ne'u 3 < Vx + 5 tiic la X > 4, bit phuong trinh da cho tuong duong vdi Vx + 5 > X + 3. Khdng cd x thoa man bit phuong trinh nay. Ne'u 3 > Vx + 5 tiic la X < 4, bit phuong trinh da cho tuong duong vdi 3 - X > Vx + 5 fx<3
fx<3
[9 - 6x + x2 > X + 5
[x2 - 7x + 4 > 0
Ket hpp ta cd : -5 < x <
7-V33 2
7-V33 ——
b) X G [-9 ; 16). c) Bit phuong trinh da cho tuong duong vdi |24-6V6-x|>-x-13.
(1)
Didu kien ciia bit phuong trinh la x < 6. * N^u -X - 13 < 0 tiic la X > -13, bit phuong trinh ludn ludn nghiem diing. vay mpi X e (-13 ; 6] la nghiem eiia bit phuong trtnh. * Vdi X < -13, ta cd V6 - x > Vl6 = 4 nen 24 - 6 V 6 - x < 0. Dodd (!)<=> 6V6 - X - 24 > - x - 1 3 o 6V6 - x > - x + l l o 36(6 - x) > x2 - 22x + 121 o x2 + 14x - 95 < 0 o -19< X < 5. Vay trong trudfng hpp dang xet, mpi x e (-19 ; -13] la nghiem ciia bit phuong trinh. Ket luan : Tap nghiem la S = (-13 ; 6] u (-19 ; -13] = (-19 ; 6]. d) X > —. Hudng ddn. Bit phuong trinh dupe vi^t thanh : |x + 3 | - | x - 3 | > 1.
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158
4.80. a) Dat r = x2 + X + 2, / > 0. Khi dd bit phuong trinh trd thinh : it-I)
it + 1)> 1 5 o r ^ > 1 6 .
(*)
Do / > 0 nen nghiem cua bit phuong trinh (*) la r > 4. Suy ra x2 + x + 2>4<=>x2 + x-2>0<=> x > l hoac x < -2. vay tap nghiem eua bit phuong trinh da cho la 5 = (-oo ; -2 } u [1 ; +oo). h)S = - 7 ; -
5+vr7
Vi7-5 ; 2
w
Hudng ddn. Dat t = ^Jx'^+5x + 2>0.
c) S = (-00 ; -2] u [6 ; +oo). Hudng ddn. Dat / = V2x2 - 8x + 12 > 0. 4.81. a) Bit phuong trinh tuong duong vdi (x - 3) >/x2 + 4 - (X + 3) <0. TCfdd tap nghiem cin tim la hpp cae tap nghiem eua hai he bit phuong trinh sau : x-3<0
x-3>0 (I)
(II) Vx2 + 4 < X + 3
Vx2 + 4 > x + 3.(*)
rx>3 Giai he (I): (I) o
, , x*^ + 4 < x2 + 6x + 9
'x>3 5 <:>X> 3. o •!
(1)
Giai he (II): Ta xet hai trudfng hpp : - Trudng hpp x < - 3 : Dd thiy mpi x < -3 la nghiem. - Trudng hpp x > -3 : Ta cd 7 ? 5 (*) o X + 4 > X + 6x + 9 <=> X < - - . vay trong trudng hpp niy, he (II) 6
cd nghiem la -3 < x < -—. (2)
Dodd(II)ox<--. Til (1) va (2) suy ra tap nghiem ciia bit phuong trtnh da cho li : /
5=
—00;
V
51 w dj
—
[3 ; +00).
booktoan.com 159
h) S =
1 1
L 3'
V3J
u
1
IV5'2
. Hudng ddn. Bit phuong trinh tuong
duong vdi he 9x2 - 4 < ^3^ ^ 2)^5x2 - 1 5x2 _ 1 > 0. 4.82. Vdi m < 0 : Phuong trinh v6 nghiem. Vdi m = 0 : Phuong trinh cd ba nghiem x = 0 ; x = ±2. Vdi m > 0 : Phuong trinh tuong duong vdi Ix2|-2|xi + m2 =0.
(1) (2)
Xet phuong trinh y2 - 2y + m2 = 0 cd A' = 1 - m2 - Ndu m > 1 tW (2) v6 nghiem nen (1) v6 nghiem. - Ne'u m = 1 thi (2) cd nghiem y = 1 nen (1) cd hai nghiem x = ±1. - Ne'u 0 < m < 1 thi (2) cd hai nghiem duong yj - 1 + VI - m2 , y2 =
i-J^ m
suy ra (1) cd bdn nghiem phan biet x, 2 = ±(1 + Vl - m ) X3^4 = ±(1 -
yjx-m^).
4.83. a) Do 11 < Vl23 < 12 va 6 < V37 < 7 nen -12 < - Vl23 < -11 va /:;:; ^ c. 9 3-Vl23 ^ . 5 2-V37 4 -7 < -V37 < -6. Suy ra - - < < -2 v a - - < ~ < --. .7^ -, 5 ^ ^^ 2 - V 3 7 3-Vi23 Vi-2<--,dod6 > . b)Tacd
tVs
= | > ^ +VTO va V35 < 6, Vl0<3,2. 5
Suy ra |V35 + VlO < ^
+ 3,2 = 6,8 < 6,9.
booktoan.com 160
4.84. Ta ed \ab - c\ = \ab - a + a - c\< \ab - a\ + \a -c\ = \a\\b-l\ + \a-c\
< 1.10+10 = 20.
4.85. a) Bit ding thiic cin chiing minh dupe bien ddi thanh : a^ + b^ + 64> \2a^b^ Ap dung bit dang thiic giUa trung binh edng va trung binh nhan, ta cd : a^ +b^ +64> 3 ^ a V . 6 4 = 12^2^3 6
L9
vay a^ + b^ + 64> 12aV hay — — ^ > 3a^b^ - 16. Dang thiic xay ra khi va ehi khi a = 2, b = \i4 b) Bit ding thiic cin chiing minh dupe bie'n doi thanh :
ia-bf + ib-4^f + ia-4bf> 0. Dau bang xay ra khi va ehi khi a = 6 = 0 hoac a = b= \ Didu nay ludn luOn diing. 4.86. a) Ta ed A =ia-\f = ia-\f ia-\)
Diu bang xay ra khi
+ ib - \f + ab - a - b + 2004 + ib+
l)2 + ( a - l ) ( ^ - l ) + 2003
- 1 -a
a-l +^
+ | ( f o - 1 ) 2 +2003 > 2003.
2
=0
-^
\a = \
vay A nho nhit bing 2003 khi o - ^ = 1. b) 5 - (fl - ^ + 1)2 + (6 - 1) 2 - 1 4 > - 1 4 . vay B nho nhit bang - 1 4 khi a = 0 , ZJ = 1. 4.87. a) Do a, />. c > 0 nen a + 6 + c > 3 ^fabc vaa^ + b'^ + c^ >3
^faVc^
Suy Taia + b + c)ia^ + b^ + c^) > 9 V « V ? = 9abc. Diu bing xay ra khi va chi khi a = b = c.
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161
b) Ap dung bit dang thUc giira trung binh epng va trung binh nhan, ta cd afe /jc — + — >2b\ c a
ac <3i7 be ac ^^ —- + — >2a ; — + —->2c, b e a b
nen
be ac ab — + -;- + — > a + b + c . a b c Dang thu'c xay ra khi a = b = c. c)
2
Dodd
a +b ^ > c. + -—— a +b
b +c b' a +c ^ , b + c + —:— > a ; a + c + — - - > b a
a b +c + a +c
+ a +b
>
a +b +c 2 ,2
Mat khae tijt bit ding thiie (x + y) > 4xy va x, y > 0 ta suy ra : 2ab a+b
a + b 2bc 2 'b+c
b +c 2ca 2 'c+a
c+a 2
Cpng timg ve' cae bit ding thiie va ehia hai ve' eho 2 ta dupe ab be ca a +b +c + < + a +b b +c ' c +a ~ 2 Ding thdc xay ra khi a = b = c. 4.88. a) Ta cd thd vie't P - |x + l| + |2x + 5| + |18 - 3x| > |(x + 1) + (2x + 5) + (18 - 3x)| = 24 (Ap dung bat ding thirc l^l + |^| + |c| > |a + ^ + c|). x + 1 >0 Diu bing xay ra khi va ehi khi (I)
x + 1 <0
2x + 5 > 0 hoac (II) • 2x + 5 < 0 18-3x>0
I8-3x<0.
He (I) cd nghiem - I < x < 6 ; He (II) vd nghiem. vay gia tri nho nhit ciia P la 24 khi - 1 < x < 6. b) Ap dung bit ding thiie
|(3: - /?| > |a| - \b\ ta dupe |x-l| > |x|-l, |y-2|>|y|-2, |z-3|>|z|-3.
booktoan.com 162
n-BTDSiaNC-B
Do dd e > |x| + \y\ + \z\ - 6 - 2006 -6 = 2000. Diu bang xay ra khi va chi khi x - 1 > 0 ; y - 2 > 0 ; z - 3 > 0 vi X + y + z = 2006. Chang han x = 2000 ; 3; = z = 3 thi|x - l| = 1999; |y - 2| = 1; |z - 3| = 0. vay gia tri nhd nhit cua Q \h 2000. f
4.89. a) 5 -
—00 :
5yl3-l^
b)S = \ ^ ; + ^
'3(V3-1) c) Bit phuong trinh dupe dua vd dudi dang ( 1 + V 3 ) X < ( 1 + yfdf
<=> x < 1 + V 3 .
V a y 5 = (-oo; 1 + V 3 ] d) Bit phuong trinh da Cho tuong duong vdi 10>(X+ V5)2 - ( x - V5)2 <;>x<
vay 5 =
f -nn
V
^5
V^" 2
4.90. a) Vdi m = 3, tap nghiem ciia bit phuong trinh la 0 Vdi m < 3, tap nghiem cua bit phuong trinh la
\
1 + m' ' m- 3 '
—00 :
Vdi m > 3, tap nghiem eua bat phuong trinh la
; + 00
V m- 3 b) Vdi m = 0 hoac m = 2, tap nghiem bit phuong trinh la E. 1 — ; + oo Vdi m < 0 hoac m > 2, tap nghiem eiia bit phuong trinh la m Vdi 0 < m < 2, tap nghiem eiia bit phuong trinh la
—00 :
m c) Ne'u m < 10 thi tap nghiem eua bit phuong trinh la -00 ; Ne'u m > 10 thi tap nghiem cua bit phuong trinh la |
m-1 m~lOj'
-^ ; +00 | ; m-10
Ne'u m= 10 thi bit phuong trinh vd nghiem. booktoan.com 163
d) Ne'u me (-QO ; - VS ] W [ VS ; +00) thi tap nghiem ciia bit phuong trinh la (-00 ; -m -Vm2 - 5 ] U [ -m + Vm2 - 5 ; +00). Ne'u me ( - V5 ; V5) thi tap nghiem eiia bit phuong trinh la R. e) Ne'u m = 0 thi tap nghiem cua bit phuong trinh la
-00 ; - —
Ne'u m > 4 thi bit phuong trinh vd nghiem. Ne'u 0 < m < 4 thi tap nghiem ciia bit phuong trinh la -2 + V 4 ^ m
-2-^4-m
m
m
Neu m < 0 thi tap nghiem ciia bit phuong trinh la
-2-V4^ m
-00:
\j
-2 +V4^ m
m
;+oo
m
f) Ne'u m = 3 thi tap nghiem cua bit phuong trinh la | -— ; + 00 Neu m < 3 thi tap nghiem ciia bit phuong trinh la -00
:
m + 1 - V3m2 - 7m + 10 u m -^ 3
m + 1 + V3m2~7m + 10 ;+oo m- 3
Ne'u m > 3 thi tap nghiem ciia bit phuong trinh la m + 1 - V3m2 - 7m + 10 m- 3
m + 1 + V3m2^^7m + 10 m-3
4.91. a) { 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 1 0 ; 11}, b) Khdng ed nghiem nguyen.
64 [33
4.92. a) m e —
N
+ 00
/
b) me '.
booktoan.com 164
4.93. a) Vd nghiem. b) S= [0 ; +00).
c)5 = ( - o o ; - 3 ) u ( - l ; 4 ) u
7 + V57
;+co .
4.94. a)xG ( - c o ; ^ ] i ^ [ - 2 ; - l ] u [ l ;+oo).//M'o?!^^i/i. Dat ?-x2 + 3 x - 1. b) X G (-2 ; -1) w (2 ; 3). Hudng ddn. Datt =
x^-x-4
c) X G (-00 ; - 2 ) u ( - 1 ; 3) LJ (4 ; +oo). d ) x G ( - 2 ; 1).
4.95. a) X e
e) X
7-V109 7 + VT09 10 10
b)XG
4 . 5^ 3'2
d)xe [; ;6J.
2
^2;
1
'
4.96. a) Do 2x - 2x + 3 > 0 vdi mpi x nen bat phuong t^inh tuong duong vdi; x2 - ( 2 + m)x + 4 > 0. Dd bit phuong trinh nghiem diing vdi mpi x, didu kien cin va du la A - (2 + m)2 - 1 6 < 0 hay - 6 < m < 2. b)m G ( - 2 ; 4). 4.97. Dat / = x2 phuong trinh trd thanh fit) = im + 3)t^ - i2m - \)t - 3 = 0, t>0. 3 • Neu m + 3 - 0, tire la m == - 3 thi /(f) = 7r - 3 = 0, ttr dd r = - . Suy ra phuong trinh da eho ed hai nghiem x - ± J-;^• Ne'u m +
3^0<^m^-3
Khi dd, A = (2m - 1)2 + I2(m + 3) = 4m2 + 8m + 37 > 0 vdi mpi m nen phuong trinh/(?) = 0 lu6n cd hai nghiem phan biet khae 0 (vi c = -3^
0).
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+) Phuong trinh/(/) =^ 0 ed hai nghiem duong khi va ehi k h i : S =
^!^>Q - 1< 0 m+3 „ f2m , . <=> <^m < - 3. m+3
Khi dd phuong trinh da cho ed bdn nghiem phan biet. +) Phuong trinh fit) = 0 ed hai nghiem am khi va ehi k h i : S= ^ ^ < 0 m +3
2m - 1 > 0
P = ^I^>0 m +3
l'" + 3 < 0
1 m > — 2 (khdng tdn tai m). m < -3
+) Phuong trinh/(r) = 0 cd mpt nghiem am va mpt nghiem duong khi va chi khi ac = (-3)(m + 3) < 0 <» m > - 3 . Khi dd phuongtrinh da cho ed hai nghiem phan biet. Tom lai:
Vdi m > ~3 phuong trinh ed hai nghiem phan biet. Vdi nh<-3 phuong trinh ed bdn nghiem phan biet.
4.98. a) Ne'u dat/(x) =^^^ - 2 thi ox + 5
/(x)>OoxG I - y ; - fix) < 0 O X G -co ; -
b) Ndu dat gix) =
x^ - 5x + 4
14 w
- - ; + 00 .
thi
X + 5x + 4 g(x)<0<^XG(-4;-l)u(l;4), ^(x) > 0 <:> X e (-00 ; - 4) u (-1 ; 1) c) Ne'u dat hix) =
15x^ - 7x - 2
L;
thi
6x - X + 5 hix) > 0 <=> X
-co;--|w
hix)<0<^xe
|-|;||.
booktoan.com 166
3'^"
(4 ; +QO).
d) Ne'u dat pix) -
x" - 17x2 ^ gQ
thi pix) > 0 khi va chi khi
x(x^ - 8x + 5) x G ( - V i 2 , - V 5 ) w ( 0 ; 4 - % / T T ) u ( V 5 ; V l 2 ) u ( 4 + VrT ;+oo). pix) < 0 khi va chi khi XG ( - o o ; - V i 2 ) u ( - V 5 ; 0 ) u ( 4 - V i T ; Vs ) u (Vi2 ; 4 + V i T ) . 4.99. a) X e (5 ; +oo); b ) X G ( - O O ; ^ ) i^ ( ^
; + oo) .
Hudng ddn. Vx^ - 4x^ + 4 =\x-^\ e)xe(-2;-l]u
4.100. a) X e 3;
6 + Vl2
ix^ + ^
+ V4).
2. P 3 '3 . Hudng ddn. Phuong trinh vie't thanh
V x - l > Vx - 2 + Vx - 3 Vdi dieu kien x > 3, binh phuong hai ve' ta dupe bat phuong trinh tuong duong X - 1 > 2x - 5 + 2^ix - 2)(x - 3) <^ 4 - X > 2V(x - 2)(x - 3) 4 - x >0 <=>
(4 - x)2 > 4(x - 2)(x - 3).
b) X G (-co, 0) w ( l , +CG). Hudng ddn. Dat t = 4x^ - x + 1 > 0 . Bit phuong trinh trd thanh 2f - / - 1 > 0. c) X > 1. d) Vie't bit phuong trinh vd dang : ( . - l ) ( x . l ) ^ ^ ^ x - l X
V
X
X
^ ( V ^ T T -1) > x - 1
Didu kien : - 1 < x < 0 hoac x > 1.
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Nhan thiy x = 1 khdng phai la nghiem eiia bit phuong trinh nen cd thd c o l X ?^ 1.
Khi dd
> 0 nen bit phuang trinh da cho tuong duong vdi
Vx + 1 - I > J
<^ Vx + 1 > 1 + J-
(*)
+ Neu - 1 < X < 0 thi Vx + 1 < 1 suy ra bat phuong trinh khdng c6 nghiem trong nua khoang [-1 ; 0). + Vdi X > 1, binh phuong hai ve' cua (*) ta di den :
( x - l ) .X i > 2V JX^ Mat khae, theo bit dang thiic Cd-si ta cd
(x-l) + Xi ^ 2V JX ^ Da'u bing xay ra khi va chi khi x - 1 = — tde la khi va chi khi X
i + Vs •I/-
,
IX
1
n l-^-l
vay (x - 1) + — > 2, X
V
1
1 + V5
<^ I < X =^ ~~— X
2
Ke't ludn. Tap nghiem cua ba't phuong trinh la 1;
i + Vs^i 2
Ti + Vs
Ju \
2
; + 00
4.101, a) X G (-00 ; - Vd) u (0 ; V2) w (2 + 2V2, + 00). b ) x G ^ - - ; + coJ. c) X e [3 ; 4) w (4, + 00). d) X G ( 1 ; 3 ) . 4.102. a) X
<-.
b) X G ( - 1 ; V 4 ) . 168
booktoan.com
c) Dieu kien |x + 3| 9^ l o
x + 3 ^ l va x + 3 ^ - 1 hay x ^-2
va
X 7^ - 4 .
* Ne'u X < - 3 , bit phuong trinh da cho tuong duong vdi ^ >-x-2 -x-3-1 <^
o ^ — < X + 2 <^ ^ — - (x + 2) < 0 x+4 x+4
3 - (x2 + 6x + 8) ^ ^ -x2 - 6x - 5 ^ ^ x2 + 6x + 5 ^ ^ ^ : < 0 <^ < 0 >0 x+4 x+4 x+4
< ^ x e [-5 ; -4). * Ndu -3 < X < - 2 , bat phuong trinh da eho tuong duong vdi ^ > - x - 2 o ^ — + x + 2 > 0 ^ 3 + (x + 2) > Q ^ . ^ > _ 2 ^ x+2 x+2 x+2 Khdng cd x thoa man yeu eiu didu kien -3 < x < -2. * Neu X > - 2 , bat phuong trinh da eho tuong duong vdi > X + 2 <» - ^ x+2
- (x + 2) > 0 <^ 3 - (x + 2)2 > 0
x+2
<^ ( V 3 - x - 2 ) ( V 3 + x + 2 ) > 0 o - 2 - V 3 < x < 2 - V 3 . vay - 2 < x < 2 - V 3 . Ket luan. x e [-5 ; - 4) u (-2 ; 2 - V3]. d) Neu X < 2 bit phuong trinh da eho tuong duong vdi > -x + 2. <^ 9 + X - 2„ .>„0 <=> 5 - x 2 + 4 x > 0 < ^ x < - l . 5 - X9 - 3 . 2-x 2-x Ne'u 2 < X < 5 bat phuong trinh da cho tuong duong vdi 9 , , _ 2 « ^ . 2 - . > 0 « ^ l i ^ ^ ^ > 0 . 5-X-3 2-x 2-x vay 2 < X < 5. Neu X > 5 bit phucrtig trinh da cho tuong duong vdi 9 9 9 > x-2 ^ > X 2 o x-5-3 x-8 X - - (x - 2) > 0 ^
9 - U ^ - 1 0 . ^ 1 6 ) ^ ^ ^ - . ^ ^10.-7 ^ ^ ^ / - l O x +7 X-8 x-8 x-8 booktoan.com 169
vay 8 < X < 5 + Vl8. Ket luan x e (-00 ; - 1] w (2 ; 5) ^ (8 ; 5 + V18]. 4.103. a) Vdi m = V5 phuong trinh trd thanh - 3 V 5 x + V5 + 1 - 0 , u-^ x = I + j=— V5 eo' nghiem 3V5 Vdi m^ 45 phuong trinh cd nghiem khi va chi khi A = 9m2 - 4(m + l)(m -VS) > 0 <=> 5m2 -4(1 -V5)m + 4V5 > 0, bit phuong trinh nay nghiem diing vdi mpi m (vi A'^ = 4(1 - V5 ) - 20 V5 < 0). vay phuong trinh da eho ed nghiem vdi mpi m. b)m e ( - l ;V5). 4.104. m < 0 : phuong trinh vo nghiem m = 0 : phuong trinh cd hai nghiem 0 < m < 4 : phuong trinh cd bdn nghiem m = 4 : phuong trinh ed ba nghiem m > 4 : phuong trinh cd hai nghiem. 4.105. Khi m = 0, dd thiy phuong trinh da cho cd mpt nghiem duy nhit x = 0. Gia six m ^ 0. Dat r = 1 - mx , ta cd x va ta duoe phuong trinh m
m|r| = t^+ (2m - 3)f + 2 - m.
. r
o
(1)
Hien nhien phuong trinh da eho ed mpt nghiem duy nhit khi va chi khi (1) cd mpt nghiem duy nhit. Ta ed phuong trinh (1) tuong duong vdi \t>0 (I) 2 h^^*^ W [t^ +im-3)t + 2-m = 0 Ta xet eac trudng hpp sau
it <0 2 [t^ + (3rn-3)t+ 2 - m = 0.
• Trudng hpp m > 2. Liie nay mdi phuong trinh bae hai trong he (I) va (II) ddu ed hai nghiem trai diu, suy ra mdi he (I) va (II) ddu cd m6t nghiem, nghia la phuong trinh (1) cd hai nghiem (trai da'u). Vay m > 2 khong thoa man didu kien eua bai toan.
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170
• Trudng hpp m < 2. Liic nay phuong trinh bac hai trong he (I) cd hai nghiem t] = I va t2 = 2 - m. Do m < 2 nen ea hai nghiem nay ddu thoa man didu kien t > 0. Vay neu r, ^ t2, tiic \a m ^ \ thi he (I) cd hai nghiem phan biet, tiic la (1) cd ft nhit hai nghiem phan biet, khOng thoa man yeu eiu ciia bai toan. Cudi ciing, khi m = 1, dd thiy he (I) cd mpt nghiem duy nhat ? = 1, he (II) vd nghiem nen phuong trinh (1) cd mpt nghiem duy nhit. Tom lai, cac gia tri cua m thoa man yeu eiu dd bai la m G {0 ; 1}. 4.106. a) Dung. b), e), d), e), f), g), h), i), k) sai. Hpc sinh tu liy phan vi du. 4.107. Phuong an (B). 4.108. Phuong an (B). 4.109. Phuong an (A). 4.110. Phuong an (D). 4.111. Phuong an (B)4.112. Phuong an (B). 4.113. Phuong an (C). 4.114. Phuong an (B). 4.115. a) o (4); b) <^ (1) ; c) o (3) ; d) <^ (2).
4.116. a) m [<1 2 ;
b) m IH 2 ;
c) m [>] 2 ;
d) m m 2.
booktoan.com 171
^huomg V THONG KE
A. NHONG KIEN THQC CAN NHO • Mpt ddu hieu \a mpt vin dd nao dd ma ngudi didu tra quan tam. Mdi ddi tupng didu tra gpi la mdt dan vi dieu tra. Mdi don vi didu tra tuong ung vdi mpt sd lieu gpi la gid tri cua ddu hieu tren don vi didu tra dd. • M6t tap eon hiru han cac don vi didu tra gpi la mpt mdu. Tap hop cae sd lieu thu dupe sau khi didu tra tren miu gpi la mpt mdu sdlieu. • Bang phan bd tin sd gdm hai ddng (hoac hai cpt). Ddng (cdt) diu ghi cac gia tri khae nhau eiia miu sd lieu. Ddng (cdt) thu' hai ghi tin sd (sd lin xuit hien eua mdi gia tri trong miu sd lieu) tuong ung. Ne'u b6 sung mpt ddng (cpt) thii ba ghi tin suit (ti sd % giiia tin sd va kich thude miu) thi ta ed bang phan bd tin sd - tin suit, • Khi sd lieu dupe ghep thanh Idp, mdi Idp gdm cac sd lieu, nim trong mpt doan (hay niia khoang) nao do, ta cd bang phan bd tin sd (tin sd tin suit) ghep Idp. • Sd trung binh dupe tinh bdi cdng thiic 1 ^ i-\
• Phuong sai dupe tinh bdi cong thiic s' =
1 J—,
1
/-I
Dp lech chuin s la can bac hai eiia phuong sai.
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172
Ne'u miu sd lieu dupe cho dudi dang mpt bang phan bd tin sd thi 1 "" ;-l 1
m
1
m
trong dd «,- la tin sd eua sd lieu x, ii = 1, 2, ..., m), ^ « j == N. (=1
• Ne'u miu sd lieu dupe eho dudi dang bang tin sd ghep Idp (Bang 7a, 7b SGK)thi N 1 m M
j-l
if'"
^2
A^' \.i=l
trong dd rt, la tin sd eiia Idp thii /, x, la gia tri dai dien eua Idp thii /, tiic li trung didm eiia doan (hay nira khoang) ung vdi Idp thir i ii = 1, 2,..., m). iV + 1 ciia miu sd lieu neu A^ le va la trung Sd trung vi la gia tri thd N N binh cpng eiia gia tri thu: ^ va ~ + 1 ne'u N chan (khi xdp cae gia tri eua miu sd lieu theo thii tu tang din). Sd trung vi dupe ki hieu la M^. • Mdt la gia tri ed tin so Idn nhit. Mdt dupe ki hieu la M^.
B. DE BAI §1. MOT VAX KHAI NI£M M 6 DAU
§2. TRINH BAY MOT MAU S 6 LI£U Trong eac bai tap cua chuong nay, yeu eiu tinh toan eac sd lieu lam tron de'n hang phin tram. 5.1. Gia ban eua 80 16 dit (don vi trieu ddng) dupe ghi lai trong bang phan bd tin sd ghep Idp sau
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Ldp
Tin sd
[79,5 ; 84,5)
5 10
[84,5 ; 89,5) [89,5 ; 94,5)
15 26 13 7 4
[94,5 ; 99,5) [99,5 ; 104,5) [104,5; 109,5) [109,5; 114,5)
a) Bo sung them cpt tin suit; b) Ve bidu do tin sd hinh cpt; e) Ve dudng gip khiic tan sd. 5.2. Didu tra vd sd tidn mua sach trong mpt nam ciia 40 sinh vien ta ed miu sd lieu sau (don vi : nghin ddng) 203
37
141
43
55
303
252
758
321
123
425
27
72
87
215
358
521
863
284
279
608
302
703
68
149
327
127
125
489
234
498
968
350
57
75
503
712
440
185
404.
a) Hay lap bang phan bd tin sd - tin suit ghep ldp gom 10 ldp. Ldp diu tien la doan [0 ; 99], ldp thiihai la doan [100 ; 199],.... iDo dai mdi doan la 99). b) Hoi cd bao nhieu phan tram sd sinh vien mua sach tiJt 5(X) ngan dong trd len ? c) Xet tdp 30% sd sinh vien diing nhidu tidn dd mua sach nhit. NgUdi mua It nha't trong nhdm nay mua he't bao nhieu ? 5.3. Vdi mdi tinh, ngudi ta ghi lai sd phin tram nhUng tre em mdi sinh ed trong lupng dudi 2500 g. Sau day la ket qua khao sat d 43 tinh (don vi:%). 5,1
5,2
5,2
5,8
6,4
7,3
6,5
6,9
6,6
7,6
8,6
6,5
6,8
5,2
5,1
6,0
4,6
6,9
7,4
7,7
7,0
6,7
6,4
7,4
6.9
5,4
7,0
7,9
8,6
8,1
7,6
7,1
7,9
8,0
8,7
5,9
5,2
6,8
7,7
7,1
6,2
5,4
7,4.
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a) Hay lap bang phan bd tin sd tan suit ghep ldp gdm 5 ldp. Ldp thd nhit la nua khoang [4,5 ; 5,5), ldp thii hai la [5,5 ; 6,5) (Dp dai mdi niia khoang la 1). b)Ve bidu dd tin sd hinh cdt. c) Ve bieu dd tin suit hinh quat. 5.4. Doanh thu eiia 19 cdng ti trong nam vita qua dupe cho nhu sau (don vi : trieu ddng) 17638
16162
18746
16602
17357
15420
19630
18969
17301
18322
18870
17679
18101
16598
20275
19902
17733
18405
18739
a) Lap bang phan bd tin sd tin suit ghep ldp, sii dung sau ldp [15 000 ; 16 000) ; [16 000 ; 17 000) [20 000 ; 21 000). b)Ve bieu dd tin sd hinh cpt; c)Ve dudng gip khiie tin sd. 5.5. Ket qua ciia mdt ki thi mdn Tieng Anh cua 32 hpc sinh dupe cho trong miu sd lieu sau (thang diem 100): 68 79 65 86
52 61 55 65
49 42 68 66
56 57 65 72
69 60 50 63
74 88 78 95
41 87 61 72
59 47 90 74.
a) Lap bang phan bd tin sd - tin suit ghep ldp, su dung sau ldp : [40 ; 50); [50 ; 60); ... ; [90; 100). b) Ve bidu dd tin sd hinh cpt. c) Ve bidu do tin suit hinh cpt.
§3 cAc sd DAC TRLTNG CUA MAU S 6 Lifiu Trong cac bai tap dudi day, yeu eiu tinh cac sd trung binh, sd trung vi, phuong sai, dp lech chuan (chinh xac den hang phin tram). 5.6. Doanh thu eua 8 ciia hang an trong mpt ngay d khu phd A (don vi trieu ddng) nhu sau : 2 2 25 2 10 100 2 10. Tim sd trung binh va sd trung vi. Sd nao lam dai dien tdt hon ?
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5.7. Mdt eau lac bd thieu nhi trong dip he cd md 7 ldp ngoai khoa. Si sd eiia cac ldp tuong ung la : 43
41
52
13
21
39
46.
Tim sd trung binh va sd trung vi. 5.8. Gia ban cua 60 mat hang d mpt eua hang dupe thdng ke trong bang tin sd ghep ldp sau day (don vi: nghin ddng). Ldp
Tin sd
[40 ; 49]
3
[50 ; 59]
6
[60 ; 69]
19
[70 ; 79]
23
[80-; 89]
•9
A^-60
Tinh sd trung binh. 5.9. Tudi cac hpe vien eiia mpt ldp hpe tie'ng Anh buoi tdi d mpt trung tam dupe ghi lai trong bang tin sd ghep ldp sau Ldp
Tan sd
[15 ; 19]
10
[20 ; 24]
12
[25 ; 29]
14
[30 ; 34]
9
[35 ; 39]
5
Tinh sd trung binh. 5.10. Nghien cun.mu'c tieu thu xang cua mpt loai 6 td, mpt cong ti che' tao 6 td d Mi da cho 35 xe chay thii va xac dinh xem vdi 1 galdng xang (Igaldng - 4,546 lit), mpt xe chay dupe bao nhieu dam (1 dam = 1,609 km). Kdt qua dupe cho trong bang tin sd ghep ldp sau day. 176
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Tan sd 2 7 15 8 3
Ldp [20 ; 24] [25 ; 29] [30 ; 34] [35 ; 39] [40 ; 44]
Tinh sd trung binh va dp lech chuan. 5.11. Sd tidn cudc phi dien thoai (don vi : nghin ddng) cua 7 gia dinh trong khu phd A phai tra dupe ghi lai nhu sau : 83 79 92 71 69 83 74. Tinh sd trung binh, sd trung vi, mdt va dp lech chuin. 5.12. Thdi didm ma cdc nhan vien cua mpt cdng ti ngu day mdi bu6i sang dupe thdng ke trong bang phan bd tin suit ghep Idp sau (don vi : gid) : Ldp [4; 5] [ 5 ; 6] [6; 7] [7 ; 8]
Tin suit (%) 7 65 24 4
Tmh sd trung binh. 5.13. Sd ngudi cip eiiu de'n benh vien A trong hai ngay thii hai va thii sau dupe cho trong bang tin sd ghep ldp dudi day. Ldp
Tan sd (trong ngay thii hai)
Tan sd (trong ngay thii sau)
[ 4 ; 7]
1
1
[ 8 ; 11]
4
4
[12; 15]
15 26
21 13
[24 ; 27]
16 7
[28; 31]
3
0
A^ = 72
^ = 64
[16; 19] [20 ; 23]
22 3
Tinh sd trung binh va dp lech chuin cua hai miu sd lieu va so sanh dd phan tan ciia ehiing.
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12-BTDSlO.NC-A
177
5.14. Sd tien dien phai tra ciia 50 hp trong khu phd A dupe thdng ke trong bang phan bd tin sd sau day (don vi nghin ddng). Ldp [375 ; 449]
[450 ; 524]
Tin sd 6 15
[525 ; 599]
10
[600 ; 674]
6
[675 ; 749]
9
[750 ; 824]
4 A^ = 50
Tinh sd trung binh va dp lech chuan. 5.15. Trong mdt dd tai nghien eun vd benh A, ngudi ta ghi lai tudi eiia 50 benh nhan mic benh nay. Sd lieu thdng ke dupe trinh bay trong bang phan bd tin sd sau day. Tin sd
Ldp [15; 19] [20 ; 24] [25 ; 29] [30 ; 34] [35 ; 39]
10 12 14 9 5
_ N = 50
Tinh sd trung binh va dp lech chuan. 5.16. Mpt ciia hang sach thdng ke sd tidn (don vi : nghin ddng) ma 60 khach hang mua sach cr eiia hang trong mpt ngay. Sd lieu dupe ghi trong bang phan bd tin sd sau : Ldp [40 ; 49] [50 ; 59] [60 ; 69] [70 ; 79] [80 ; 89]
Tin sd 3 6 19 23 9 A^ = 60
Tinh sd trung binh va dp lech chuin. 178
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12-BTDSlO.NC-B
5.17. Ngudi ta chpn mdt sd but bi cua hai hang san xuit A va B va thd xem sii dung mpt biit sau bao nhieu gid thi het mire. Ke't qua nhu sau (don vi gid): Loai biit A-.23 25 27 28 30 35. Loai but B : 16 22 28 33 46.a) Tinh sd trung binh va dp lech chuan vd thdi gian sii dung cua mdi loai biit. b) Gia sir hai loai but A va B cd ciing mpt gia. Dua vao su khao sat tren, ta nen quyet dinh mua loai biit nao. 5.18. Khdi lupng (don vi ; pound ; 1 pound = 0,454 kg) ciia mdt nhdm ngudi tham gia eau lac bp siic khoe dupe ghi lai nhu sau : 175 166 148 183 206 190 128 147 156 166 174 158 196 120 165 189 174 148 225 192 177 154 140 180 172 135. Tinh sd trung binh, sd trung vi va mdt. 5.19. van tde (dam/h ; Idam = 1,609 km) eiia 400 xe dtd chay tren con dudng A dupe ghi lai trong bang tin sd ghep ldp sau : Ldp [27,5 ; 32,5) [32,5 ; 37,5) [37,5 ; 42,5) [42,5 ; 47,5) [47,5 ; 52,5)
Tan sd 18 76 200 100 6^ A'-400
Tinh sd trung binh, do lech chuan. 5.20. Mpt eiia hang an ghi lai sd tidn (nghin ddng) ma mdi khach tra cho eua hang. Cac sd lieu dupe trinh bay trong bang tin sd ghep ldp sau : Ldp [0; 99] [100; 199] [200 ; 299] [300 ; 399] [400 ; 499]
Tin sd 20 80 70 30 10 iV-210
Tinh 'sd trung binh va dp lechbooktoan.com chuin. 179
5.21. Diem trung binh thi hpe ki mdn Toan eua hpe sinh nam va nU eiia hai trudng AvkB cung nhu ciia mdi trudng dupe thdng ke trong bang sau : Trudng A B AvhB
Nam 7,18,1 7,9
Nam va nfl 7,4 8,4
Nii 7,6 9,0
Tinh didm trung binh ciia hpc sinh ca hai trudng A vh B (chinh xac tdi hang phan ehue).
BAI TAP 6 N TAP CHUCfNG V 5.22. Chidu cao eiia mpt miu gdm 120 cay dupe trinh bay trong bang phan bd tin sd ghep ldp sau day (don vi ; m) : Ldp
Tin sd
[1,7; 1,9)
4
[1,9; 2,1)
11
[2,1 ;2,3)
26
[2,3 ; 2,5)
21
[2,5 ; 2,7)
17
[2,7 ; 2,9)
11
[2,9; 3,1) [3,1 ; 3,3)
7 6
[3,3 ; 3,5)
7
[3,5 ; 3,7)
3
[3,7 ; 3,9)
5
[3,9; 4,1)
2 N=\20
a) Ve bidu dd tin sd hinh cpt. b) Ve dudng gip khiie tin sd. c) Dua tren hai bidu dd nay, cd nhan xet gi ve xu the phan bd chidu cao cua cay ? Phan Idn sd cay cd chidu cao nim trong khoang nao ? .
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5.23. Trong ta't ea cac miu sd lieu kich thude 5 vdi so trung vi la 12 va sd trung binh la 10, hay tim mpt miu sd lieu cd bien dp nhd nhit (bien dp ciia miu sd lieu la hieu giiJa gia tri Idn nhit va be nhit cua miu so lieu). 5.24. Mpt edng ti cd 45 chid'c xe. Miic tieu thu xang (don vi : lit) ciia mdi xe trong tuin qua dupe ghi lai nhu sau : 123 132 130 119 106 97
121 109 118
128 132 115 130 125 121 127' 144 115 107 110 112 118 115 134 132 139 144 104 128 138 114 121 129 128 116 138 129 113 105 142 122 131 126 111 142 a) Lap bang phan bd tin sd ghep ldp vdi cae ldp la : [90; 100), [100; 110),..., [140 ;150). b) Tinh sd trung binh va sd trung binh (xip xi) dua tren bang phan bd tin sd ghep ldp. c) Tmh sd trung vi.
Gl6l THifeU MOT S 6
CAU
HOI TRAC NGHlfiM K H A C H QUAN
Trong cdc bdi tic 5.25 de'n 5.26, hay chon phuang dn trd ldi dung trong cdc phuang dn da cho. 5.25. Bang phan bd tin sd sau day ghi lai sd ghd trdng trong eac chuydn bay tiJf Ha Ndi den Tp. Hd Chi Minh. Ldp
Tin sd
[0 ; 4]
3
[5-9]
8
[10;14]
15
[15 ; 19]
18
[20 ; 24]
12
[25 ; 29]
6
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TI le phin tram sd chuydn bay cd nhidu nhit 19 ghd trdng xip xi la (A) 15%; (B)29%; (C) 71% ; (D) Khdng thd xac dinh dupe tir bang tren. N 5.26. Gia sir kich thude miu la A^. Khi dd ludn cd N_ (phin nguyen c ^ a y ) sd 2 lieu trong miu Idn hon hoac bing (A) Sd trung binh ; (B) Sd trung vi; (C) Mdt; (D) Dp lech chuin. 5.27. Chpn khang dinh sai trong cae khang dinh sau vd sd trung binh x (A) Tit ea eac sd lieu trong miu ddu phai dung dd tinh sd trung binh x ; (B) Sd trung binh x bi anh hudng bdi eac gia tri qua Idn hay qua be ; N
iC)T6n$J^iXi-x)
= 0;
(-1
(D) Mpt nua sd lieu trong miu Idn hon hoac bang x . 5.28. Chpn khing dinh diing trong cae khing dinh sau vd sd trung vi M^ (A) Sd trung vi ludn la mpt sd lieu nao dd cua miu ; (B) Sd trung vi bi anh hudng bdi cac gia tri qua Idn hay qua bd ; N
(C)T6ngX(-^/-M,)-0. N~\ (D) Cd — sd lieu Idn hon hoac bang M^, d dd TV la kich thude mau.
Trong cdc bdi tii; 5.29 den 5.33, hay chon phuang dn trd ldi dung trong cdc phuang dn dd cho. 5.29. Cac gia tri xuit hipn nhidu nhit trong miu sd lieu dupe gpi la (A) Mdt; (B) Sd trung binh ; (C) Sd trung vi; (D) D6 leeh chuin. 5.30. Neu don vi do cua sd lieu la kg thi don vi eua dp lech chuin la (A)kg ; (C) Khdng cd don vi (hu sd);
(B) kg^; (D) kg/2.
5.31. Mdt hpc sinh ghi lai bang phan bd tin suit ciia mpt miu sd lieu nhu sau
182
Gia triix) Tin sd
0
1
Tin suit (%)
12,5
2
3
4 A'^
0,0
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50,0
25,0
12,5
100
Tuy nhien, em dd quen ghi kich thude mau A^. Khi dd, gia tri nho nhit cd thd cd cua A^ la (A) 5 ;
(B)8;
(C)16;
(D) 25.
5.32. Cho X, Y, Z la ba miu sd lieu ddi mpt khdng cd phin ttr ehung. Sd trung binh ciia cac miu sd lieu X,Y,Z,X^ Y,Xu ZvaY u Z dupe cho trong bang dudi day. Miu
X
Y
Z
XyjY
X^Z
YuZ
Sd trung binh
37
23
41
29
39,5
33
Khi dd, sd trung binh eiia miu X u 7 u Z la (A) 33 ;
(B) 33,5 ;
(C) 33,66 ;
(D) 34.
5.33. Hpc sinh tinh A (gom ldp 11 va ldp 12) thain du ki thi hpe sinh gidi Toan ciia Tinh (thang didm 20) va didm trung binh ciia hp la 10. Bie't rang sd hpe sinh Idp 11 nhidu hon sd hpc sinh ldp 12 la 50% va didm trung binh ciia khdi 12 cao hon didm trung binh eiia khdi 11 la 50%. Didm trung binh ciia khdi 121a (A) 10;
(B) 11,25 ;
(C) 12,5 ;
(D) 15.
C. DAP SO - HUONG DAN - LOI GlAl 5.1. a) Bd sung cpt tin suit, ta dupe Ldp
Gia tri dai dien
Tin sd
Tin suit (%)
[79,5 ; 84,5)
82
5
6,25
[84,5 ; 89,5)
87
10
12,50
[89,5 ; 94,5)
92
15
18,75
[94,5 ; 99,5)
97
26
32,50
[99,5 ; 104,5)
102
13
16,25
[104,5; 109,5)
107
7
8,75
1109,5 ; 114,5)
112
4
5,00
N = 80
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b) Bidu dd tin sdhinh cdt (h.5.1)
79,5 84,5 89,5 94,5
99,5 104,5 109,5 114,5
Hinh 5.1
c) Dudng ga'p khiic tin sd (h.5.2)
"^n
M4
25' ?0' 15'
Ms/
10' 5-
\M5 ^ 6
M i ^
lAi
82
1
I
!A2
IA3
87
92
.A4
97
IA, 102
lAfi
107
M7 ^A7
112
Hinh 5.2
5.2. a) Bang phan bd tin sd - tin suit ghep ldp : [ 0 ; 99]
Tin sd 9
Tin sua't(%) 22,5
[100; 199]
6
15,0
[200 ; 299]
6
15,0
[300 ; 399]
6
15,0
[400 ; 499]
5
12,5
[500 ; 599]
2
5,0
[600 ; 699]
1
2,5
[700 ; 799]
3
7,5
[800 ; 899]
1
2,5
[900 ; 999]
1
2,5
Ldp
N = 40 booktoan.com
184
b) Nhin vao bang tren, ta tinh dupe ti le sinh vien mua tir 500 ngan trd len la 5% + 2,5% + 2,5% + 2,5% + 7,5% - 20%. 30 = 12 100 sinh vien. Cd tam sinh vien tieu tir 500 ngan trd len. Ta cin chpn them bdn sinh vien nira trong nhdm thii 5, nhom tieu tidn trong doan [400 ; 499] ; nam sd lieu trong nhdm nay la 498 ; 489 ; 440 ; 425 va 404. Do do, ngudi mua it nha't la 425 nghin ddng. e) Xet tdp 30% sd sinh vien mua nhidu tidn nha't. Nhdm nay ed 40 x
5.3. a) Bang phan bd tin sd - tin suit ghep ldp
Ldp
Tin sd
Tin suit (%)
[4,5 ; 5,5)
9
20,93
[5,5 ; 6,5)
6
13,95
[6,5 ; 7,5)
17
39,53
[7,5 ; 8,5)
8
18,60
[8,5 ; 9,5)
3
6,98
N = 43 b) Bidu dd tin sd hinh cdt (h.5.3)
4,5
5,5
6,5
7,5
8,5
e) Bidu dd tan suit hinh quat (h.5.4)
9,5
Hinh 5.3
Hinh 5.4
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5.4. a) Bang phan bd tin sd - tin suit ghep ldp
Ldp
Gia tri dai dien
Tin sd
Tin suit (%)
[15000; 16000)
15 500
1
5,26
[16000; 17000)
16500
3
15,79
[17 000; 18 000)
17 500
5
26,32
[18000; 19000)
18 500
7
36,84
[19000; 20000)
19500
2
10,53
[20000; 21000)
20500
1
5,26
b) Bidu d6 tin sd hinh cpt (h.5.5)
16
17 18 19 20 (dcfn vi: nghin)
e) Dudng gip khiic tin sd (h.5.6)
15,5 16,5 17,5 18,5 19,5 20,5
Hinh 5.5
Hinh 5.6
5.5. a) Bang phan bd tin sd - tin suit ghep ldp
Ldp [40 ; 50) [50 ; 60) [60 ; 70) [70 ; 80) [80 ; 90) [90 ; 100)
Tin sd 4 6 11 6 3 2 TV = 32 booktoan.com
186
Tin sua't(%) 12,50 18,75 34,38 18,75 9,38 6,25
b) Bidu dd tin sd hinh cpt (h.5.7)
e) Bidu do tin suit hinh cdt (h.5.8) 35 30 25' 20 15 10 . H H O 5-
40
50
60
70
80
90
40
100
50
Hinh 5.7
1 ,
• 4^ 60
70
80
90
100
Hinh 5.8
5.6. Sd trung binh 19,13. Sd trung vi la 6. Sd trung vi lam dai dien tdt hon vi cd su chenh lech Idn giUa cac sd lieu trong miu. 5.7. Sd trung binh la 36,43. Sd trung vi la 41. 5.8. Sd trung binh la 69,33. 5.9. Sd trung binh la 25,7. 5.10. Sd trung binh la 32,43. Phuong sai la 24,82. Dp lech chuin la 4,98. 5.11. Sd trung binh la 78,71. Sd tmng vi la 79. Md't l i 83. Phuong sai la 55,63. D6 lech chuin la 7,46. 5.12. Tit cdng thu'c tinh sd trung binh ta thiy 1
ffi
m
m
^='^ll''i''i = llij^i = llMi1=1
i^i
i-i
Thay vao cdng thiie tren ta thu dupe sd trung binh la 5,75 gid tu'c la 5 gid 45 phiit. 5.13. Ddi vdi miu sd lieu sd ngudi ca'p cihi trong ngay thii hai : Sd trung binh la 18,43 va dp Ipch chuin la 4,73. Ddi vdi miu sd lipu sd ngudi cip eun trong ngay thii sau : Sd trung binh la 16,69 va dp lech chuin la 4,13. Dp phan tan ciia miu sd lieu sd ngudi cip ciiu trong ngay thu" sau nhd hon. 5.14. Sd trung binh la 576,1. D6 lech chuin la 113,08. 5.15. S6 trung binh la 25,7. D6 lech chuin la 6,23. 5.16. Sd trung binh la 69,33. Dd lech chuin la 10,25.
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187
5.17. Loai but A : Sd trung binh la 28 gid ; Dp lech chuin la 3,83 gid. Loai but B : Sd trung binh ia 29 gid ; Do lech chuin la 10,24 gid. Loai biit B cd thdi gian sir dung trung binh lau hon. Tuy nhien, do dd lech chuan Idn hon nen chit lupng cua but B khdng dong ddu. Ndu khdng may ban ed the mua phai chie'c but cd thdi gian sir dung rit thip. 5.18. Sd trung binh la 167,8 pound ; Sd trung vi la 169 pound. Mdt ed ba gia tri la 148 pound, 166 pound va 174 pound. 5.19. Sd trung binh la 40 dam/h. Dp leeh chuin la 4,12 dam/h. 5.20. Sd trung binh la 216,17. Dp lech chuin la 99,20. 5.21. Gpi sd hpc sinh nam trudng A\a a ; sd hpc sinh nil trudng A la a ; sd hpc sinh nam trudng B\ab ; sd hpe sinh nir trudng B la b'. Tdng sd didm. ciia hpc sinh nam trudng A la SiA) = 7,1a. Tdng sd diem cua hpc sinh nU trudng A XaS'iA) = l,6d. Tdng sd didm eua hpe sinh toan trudng A la 5(A) + S '(A) = 7,4(a + d). Suy ra 7,I<7 + 7,6a' - 1,4a + l,4d
Jvl dd 0,2a' = 0,3.3 hay d - 1,5a.
(1)
Tdng sd didm ciia hpe sinh nam trudng B la 5(S) = 8,1 b. Tdng so diem eua hpc sinh nii trudng B \aS\B)
= 9,0 b\
Tdng sd didm ciia hpc sinh toan trudng B la SiB) + S XB) = 8,4 ib + /?'). Suy ra 8,1^> + 9,0^' = S,4b + H,4b\ Tit dd 0,6^' - 0,3b hay b' - 0,5b.
(2)
Tdng sd didm ma hpe sinh nam cua hai trudng A va 6 nhan dupe la 5(A) + SiB) = 7,9(a + b). Suy ra 7,1a + 8,1/? - 7,9a + 1,9b. Tit do 0,2b = 0,8a hay b = 4a. TIT
(3)
(2) va (3) suy ra b' = 2a.
T6ng sd didm cua hpc sinh ca hai trudng A va B la §iA) + SiB) +S\A)
+ S\B) = 7,4(a + d) + 8,4(/? + h')
- 7,4a + 7,4.1,5a + 8,4.4a + 8,4.2a - 68,9a. Sdhpc sinh ca hai trudng AvaB\aa
+ d + b + b' = a+ l,5a + 4a + 2a = 8,5(3.
vay diem trung binh cua hoc sinh hai trudng la ~-— o,5a booktoan.com
« 8,11.
5.22. Ta ed Ldp
Gia tri dai dien
Tin sd
Tin suat(%)
[1,7; 1,9)
1,8
4
3,33
[1,9; 2,1)
2,0
11
9,J7
[ 2 , 1 ; 2,3)
2,2
26
21,67
[2,3 ; 2,5)
2,4
21
17,50
[2,5 ; 2,7)
2,6
17
14,17
[2,7 ; 2,9)
2,8
11
9,17
[2,9; 3,1)
3,0
7,
5,83
[ 3 , 1 ; 3,3)
3,2
6
5,00
[3,3 ; 3,5)
3,4
7
5,83
[3,5 ; 3,7)
3,6
3
2,50
[3,7 ; 3,9)
3,8
5
4,17
[3,9; 4,1)
4,0
2
1,67
N= 120
a) Bidu do tin sd hinh cpt (h.5.9)
1,7 1,9 2,1 2,3 2,5 2,7
2,9
3,1 3,3 3,5
3,7 3,9
4,1
Hinh 5.9
booktoan.com 189
b) Dudng gip khiic tin sd (h.5.10)
1,8
2,0 2,2
2,4
2,6
2,8 3,0 3,2
3,4
3,6
3.8
4,0
Hinh 5.10
c) Nhin vao bang tren ta thiy : Chidu cao ciia cay nim trong khoang tir 1,7m de'n 4,1m. Cd 53,34% sd cay cd chidu cao tiJf 2,1m de'n 2,7m va ed 88,34% sd cay cd chidu cao tir 1,9m den 3,5m. 5.23. Gia s i r a < ^ < m < c < £ i l a miu sd lieu kich thude 5 va cd sd trung binh la 10 va sd trung vi la 12. Tir gia thiet suy ram= \2vaa y\c + d>\2+n suyxa2a
= 24ntna
+ b^3^-ic
+
+ b + c + d = 3%.
d)<3%-24=\4.Via
+ b < 14. Vay a < 7. Khi dd, bien d p B - ^ - a >
12-7-5.
Miu sd lieu (7 ; 7 ; 12 ; 12 ; 12} ed sd trung binh la 10 va sd trung vi la 12 vdi bien dp 5. Dd chinh la miu sd lieu cd bien dp be nhit trong sd eac miu sd lieu kich thude 5 vdi sd trung binh 10 va sd trung vi 12. 5.24. a) Bang phan bd tin sd ghep ldp
Ldp
Gia tri dai dien
Tin sd
[90 ; 100)
95
1
[100; 110)
105
5
[110; 120)
115
12
[120; 130)
125
13
[130 ; 140)
135
10
[140; 150)
145
4 A' = 45
190
booktoan.com
b) TCr dd tinh dupe sd trung binh (tinh theo bang phan bd ghep ldp) la 123,44 ll?. Ne'u tinh diing tren miu sd lieu (khi khdng ghep ldp) thi sd trung binh la 123,11 lit. c) Di tinh sd trung vi, ta sip xep cac sd lieu tren theo thii tu tang din nhu sau : 97
104
105
106
107
109
110
111
112
113
114
115
115
115
116
118
118
119
121
121
121
122
123
125
126
127
128
128
128
129
129
130
130
131
132
132
132
134
138
138
139
142
142
144
144.
Tiic dd sd trung vi M^ = 123. 5.25. (C).
5.26. (B).
5.27. (D).
5.28. (D).
5.29. (A).
5.30. (A).
531. iB). Hudng ddn : Sd lieu ed gia tri 0 'va 4 cd tin sd la N '^^' = ^ 100 8
Do (So , A^ phai chia
N . he't chc) 8. Sd lieu cd gia trj 2 cd tin sd la A^-TT^ -• . ~ v a sd lieu ed gia
25 N tri 3 la A^-77^ ^ X
^ ^ *'°' ^ P^^^ ^^^^ ^^'^ eho 8 ; 4 ; 2, tu'c la phai
chia he't eho BCNN (bpi ehung nhd nhit) ciia 8 ; 4 ; 2. Ma BCNN eiia 8 ; 4 ; 2 la 8. Do dd, N phai chia he't cho 8. Vay gia tri nhd nhit cua A^ la 8. 5.32. (D). Hudng.ddn : Ki hieu n,mvak tuong ung la kich thude eua miu X K va Z ; SiX), 5(7) va 5(Z) tuong ung la tdng tit ca cac gia tri ciia sd lieu trong miu X, Y va Z. Theo bai ra ta cd 5(J^ = 37rt, 5(F) = 23m, 5(Z) = 4 U va SiX) + SiY) = in + m)29. Suy ra 37/1 + 23m = 29rt + 29m. Tir dd 8n = 6m hay n - 0,75m. Tuong tu, VI SiY) + SiZ) = (m + k)33 nen suy ra 23m + 4U = 33m + 33/t. Til dd U = 10m hay k = 1,25m.
booktoan.com 191
Tong tit ca cac gia tri ciia sd lieu trong miu X u Y u
Z\a
SiX) + SiY) + S{Z) = 31 n + 23m + 41^ = 37.0,75m + 23m + 41.1,25m = 102m. Kich thude ciia miu X u y u Z la n + m + k = 0,75 m + m + 1,25m = 3m. vay sd trung binh cua mau Xu
102m F u Z la --— - 34. 3m
5.33. (C). Hudng ddn : Gpi sd hpc sinh ldp 12 la n. Theo bai ra, sd hpe sinh ldp 11 se la 1,5«. Gpi diem trung binh ciia hpe sinh ldp 11 la a. Theo bai ra, diem trung binh cua hpc sinh ldp 12 la 1,5a. Tong sd diem eua hpe sinh ldp 11 la 5 = a. 1,5« = 1,5an. Tong sd didm ciia hpe sinh Idp 12 la 7 = ( 1,5a) n = 1,5an. vay tong sd didm cua hpc sinh ldp 11 va 12 la l,5a« + l,5art = 3an. Mat khae, ta cd tong sd hpc sinh ldp 11 va 12 la n + 1,5/7 = 2,5/z va diem trung binh ciia ldp 11 va 12 la 10. Do dd, t6ng sd diem ciia hpc sinh ldp 11 v a l 2 1 a 10.(2,5/1) = 25/1. 25 Td dd ta ed 3an = 25n hay a= ~~25 Vay diem trung binh cua hpc sinh ldp 12 la \,5a = 1,5.— = 12,5.
192
booktoan.com
(phuang VI GOC LUONG GIAC VA CONG THLfC LJONG GIAC
A. NHQNG KIEN THQC CAN NHO 1. Cung tron. Quan he giCra do va radian Cung tron ban kinh R c6 so do radian a iO < a < 2n), ed sd do a° (0 < a < 360), ed dp dai / thi :
¥ = ISO-'=''"• 2. Cong thurc lUdng giac cd ban ^ sin( a + k2n) = sin a tan« =
cos( a + k2K) = cos a
sin a cos a
t a n ( « + kn) = tana
cos or
1
Sin a
tana
cot(a + kn) = cota 1 cos a 1
COS ff + sin a = I
sin a 3. Gia trj lUdng giac cua cac goc (cung) co lien quan dac biet sin(-«) = -sinor
sin(7t + a) = - s i n a
sin(7r ~a) = sina
cos ( - a ) = cos or
cos(7i + a) = -cos a
cosin - a) = -cos a
tan(-«) = - t a n a
tan(;t + a) = tana
tanin - a)= - t a n «
. (n \ sm! -X- oc = c o s a
(TI \ . cos -z - cc \ = sina
13-BTDSiaNC-A
booktoan.com
(-^ \ tan — - a \= cota 193
4. Mot so' cong thiirc lirdng giac • Cdng thufc cpng cos(« + P)= COS acosp - sin a sin p eos(« - P)= cos a cos p + sinasinp sin(a + 13)= sin « cos y^ + cos a sin ^ sin(a - f3)= sin«cos>9 - cosorsin/? tzx\ia + p) =
tan a + tan y^ 1 - tan a tan y5
tan (2 - tan yff tania - J3) = 1 + tan a tan /? Cong thurc nhan dot 2
2
2
2
cos2a= cos or - sin a = 2eos a - 1 = 1 - 2sin a ; sin2<3:= 2sin«eosa ; tan2a 2tanff 1 - tan a • Cong thurc ha bac 2 1 +COS 2a . 2 1-cos 2a cos a = ; sin a= . 2 2 • Cdng thurc bien dot tich thanh tdng •J
eosaeos^= — [eos(a + p) + eos(a - p)] smacosp=
- [sin(a + p) + sin(a - p)]
sinasinyff= -—[eos(a + p) - eos(a ->9)] • Cdng thurc bien doi tong thanh tich „ a +p a-p cosa + eosp-2eos——-^eos——^— „ ^ . a + P . a- P cosa - eosp = -2sm—^-^sin — ^ 2 2
-a ^ . a + p a- p ^cos — sma + smp= 2sin ^ 2 2 . ^ ^
a +P .
a-P
sma - sinp= 2cos——^sm——^ booktoan.com 194
13-BTDS10,NC-B
B. DE BAI §1. GOC v A CUNG Ll/ONG G l A c §2. GIA TRI LLfONG GIAC CUA GOC (CUNG) 6.1.
LUONG
GIAC
Trong cae khing dinh sau day, khing dinh nao dung, khing dinh nao sai ? a) Gde lupng giac iOu, Ov) ed sd do duong thi mpi gde lupng giac cung tia diu, tia cudi vdi nd cd sd do duong. b) Goc lupng giac iOu, Ov) ed sd do duong thi mpi gde lupng giac (Ov, Ou) cd sd do am. c) Hai gde lupng giac iOu, Ov) va iOu, Ov') cd sd do khae nhau thi cac gde hinh hpe uOv, u'Ov' khdng bang nhau. IITT
1371
6
6
--^
—•—•
d) sd(Ow, Ov) = ^ , sd iOu', Ov) - —— thi uOv - u'Ov'. e) Hai gde lupng giac (0«, Ov) va iOu, Ov') c6 sd do sai khae mpt bpi nguyen ciia 2n thi cac gde hinh hpc uOv, u'Ov' bang nhau. f) Hai gde hinh hpc uOv, u'Ov' bang nhau thi sd do cua cae gde lupng giac iOu, Ov) va (OM', Ov') sai khae nhau mpt bdi nguyen eiia 2n. 6.2. D6i sd do radian ciia cung tron sang sd do dp : .371
b)
271
3 '
e) 2,3 ; 6.3.
c)
llTl
6
f)4,2
Ddi sd do dd eua cung tron sang sd do radian : 3)45**; b) 150°; c) 72° ; d) 75°
6.4. Mpt day curoa quSn quanh hai true tron tam / ban kinh 1dm va tam .f ban kinh 5dm ma khoang each / / la 8dm (h.6.1). Hay tinh dp dai ciia day.eu-roa. 6.5.
Hinh 6.1
0-ra-to-xten (Eratosthene), d the' ki thii II trudc Cdng nguyen (Nguyen giam dde thu vien n6i tieng booktoan.com d A-le'ch-xang-dri (Alexandrie)) da tim each 195
tinh ban kinh eiia Trai Dat bang each do khoang each giCra hai thanh phd A-lech-xang-dri va Xy-en (Syene) la 8004km (theo don vi ngay nay ; thud dd cae doan lac da di tCr thanh phd nay de'n thanh phd kia ma't 50 ngay dudng). Bie't rang, khi d Xy-en tia sang mat trdi ehie'u thing dirng (nhin thing xudng gieng sau), thi cf A-le'ch-xang-dri, tia sang mat trdi lam mdt gde (7,1) vdi phuong thing dutig. Hoi lam sao Hinh 6.2 O-ra-tO-xten suy ra dupe ban kinh cua Trai Dat (xa'p xi 6 400 km) (h. 6.2) ? 6.6. Banh xe may ed dudng kinh (ke ca ldp xe) 55 cm. Ne'u xe chay vdi vSn tde 40 km/h thi trong mpt giay banh xe quay dupe bao nhieu vdng ? 6.7. Xet hinh quat tron ban kinh R, gde d tam a iR>0,0< a<2n). (h. 6.3). a) Biet dien tich hinh tron ban kfnh R la nR va dien tfeh hinh quat trdn ti le thuan vdi sd do gde o tam. Hay tinh dien tich hinh quat trdn ndi trdn. Hoi a bing bao nhieu thi dien tfeh dd bang R^ ?
Hinh 6.3
b) Gpi chu vi hinh quat trdn la tong dp dai hai ban kinh va dd dai cung trdn ciia hinh quat dd. Trong cac hinh quat ed chu vi cho trudc, tim hinh quat cd dien tich Idn nha't. c) Trong cac hinh quat cd dien tich cho trudc, tim hinh 'quat cd chu vi nho nha't. 6.8. Huyen li Quang Ba tinh Ha Giang va huyen li Cai Nude tinh Ca Mau cung nim d 105° kinh dong, nhung Quang Ba d 23° vi bie, Cai Nude d vi dp 9° bie. Hay tinh dd dai cung kinh tuye'n ndi hai huyen li dd ("Khoang each theo dudng chim bay"), coi Trai Dat cd ban kinh 6378km. 6.9. Tim sd do dp cua cae cung lupng giac cd sd do radian sau ; . In
b)
-1771
c)
1371
booktoan.com 196
d)-l,72.
6.10. Diing may tinh bd tui, doi sd do dp ra sd do radian chinh xac ddn sd thap phan thii ba : a) 2 0 ° ;
b)-144°;
e) 2003°;
d) 7t°
6.11. Cho gde lupng giac (OM, OV) cd sd do— Hoi trong cac sd — ; — ; 1171 3l7r 1471 , - . ,v . , . , , ., , — — ; —— ; — — , nhung so nao la so do eua mpt goe lupng giac co Cling tia dAu, tia cudi vdi gde da eho ? 6.12. Hay tim sd do a eua gde lupng giae iOu, Ov) vdi 0 < a < 271, bie't mpt gde lupng giac ciing tia ddu, tia cudi vdi gde dd co sd do la : 2971 ^ ;
12871 3-;
200371 ^^ . ^;18,5.
6.13. Hay tim sd do a^ eua gde lupng giac (OM, OV), 0 < a < 360, bie't mpt gde lupng giac ciing tia dSu, tia cudi vdi gde dd ed s6' do la ; 3 9 5 ° ; - 1 0 5 2 ° ; - 9 7 2 ° ; (2071)°. 6.14. a) Trong cae gde lupng giac cd tia d^u Ou, tia cudi Ov cho trudc, chiing minh ring, cd mpt gde lupng giac duy nha't (Ow, Ov) cd sd do a, - 71 < a < 71 va chiing minh ring \cA la sd do radian eua gde hinh hpe MOV. b) Tim sd do cua gde hinh hpe uOv, bie't gde lupng giac (OM, OV) cd sd do la : 971
57: 1067:
....
• 220° ; - 235" ; 1945° ; -2003° 6.15. a) Chiing minh ring ne'u sd(OM, Ov) = a, sd(OM', Ov') = p thi cac gde hinh hpe uOv, u'Ov' bing nhau khi va ehi khi hoac p - a = k2n hoac
P+a = k2nike Z). b) Hoi trong cac cap gde lupng giac (OM, OV) ; (OM', Ov') cd sd do nhu sau, cap nao xae dinh cap gde hinh hpe MOV ; u'Ov' bang nhau ? 137:
. UTI
I2II7: 8
1371 ,
UT:
177: .
157:
7317: .
II7:
2003;r .
booktoan.com 197
6.16. Tren mpt dudng trdn dinh hudng cho ba diem A, M, N sao cho ^
;:
(^7:
"^
sd AM = - ; sd AN = ——, (/:e Z). Tim k G N di M trung vdi N va tim 6 /yo k G N di M va N ddi xiing qua tam dudng trdn. 6.17. Tren mpt dudng trdn dinh hudng cho ba diem A, 'M, N sao cho "^
^
T:
371
sd AM = — ; sd AN = —
Gpi P la diem thude dudng trdn dd de tam
r\ giac MNP la tam giac can. Hay tim sd do AP 6.18. Tren dudng trdn lupng giac hay tim eac dilm xac dinh bdi cac sd : ^ + k^,ikGZ);
^|.(^eZ);
k^^keZ).
6.19. Tim gia tri lupng giac sin, cosin, tang ciia cac gde lupng giac ed sd do sau (khong diing may tfnh) : • 120° ; -30° ; -225° ; 750° ; 510° 57C 7jt
57:
IOJ:
177:
7C
1
6.20. Cho sd a ,— < a < J:. Hoi cae diem tren dudng trdn lupng giac xac dinh bdi cae sd sau nim trong gde phin tu nao eua he toa dp vudng gde gin vdi dudng trdn dd : 7:
7:
37:
„
a-7r;a-i- — ; — - a ; — - - a ? 2 2 2 6.21. Xac dinh da'u cua sina, cosa, tana, biet: 37:
37:
77:
7i
T
^^
^T'
^ ^ , 27: < a < 2,57:;
~ IOT: 37: < a < - — ;
77t
.
~4~^^ ^
.
•
'
5?: II7: — < a < ~r-
6.22. Trong mat phing toa dd Oxy, xet c5c didm M cd toa dd : (3 ; - 4), (4 ; -3), (-l2;-9),(-l;l). Hay tfnh cac gia tri lupng giac cua cdc gde lupng giac iOx ; OM).
booktoan.com 198
. n
K
o
J
A = cos—sin -— + sin—cos— ;
6.23. Tfnh
r B=CQS
-|]eos
j,\
+ Sin
.4y
f
3711
sm
k2n 6.24. Hoi cd bao nhieu gia tri khae nhau cua sin——, khi sd nguyen k thay ddi ? k2u k2'7i ku CGng cau hoi dd eho cos—— ; tan—— ; t a n — 6.25. Diing may tfnh bd tui, tim cae gia tri lupng giac sau (ehfnh xac de'n hang phin nghin) : sinlO ; cos— ; tan-—- ; cot(l,35). 6.26. Tinh cae gia tri lupng giac edn lai eiia a, bie't: ^
5
. 371
_
a) cosa = 7T^^ ~^^
7:
a<2n\
3;: 15 . c) tana =-— v a 7 : < a < - - - ; 8 2 6.27. Cho tana = 3 . Tfnh
b) sina = 0,8 va — < a < TI ; 37C
d) cota = - 3 va — < a < 271.
2 s i n a -*- 3cosa
3sina - 2 c o s a
4 s i n a - 5 c o s a ' 5sin^a + 4cos^a "
6.28. Chiing minh ring : a)
tan^ a - sin a cot a - cos a
= tan a ;
, ^ sma + cosa , ^2 . 3 = 1 -I- tan a + tan a + tan a ; b) — : — r cos a :) yjsm^a (1 + cota) + cos a ( 1 -i- tan a) = |sin a + cos a\; 0
0
0
0
2
d) sin a tan a + 4sin a - t a n a + 3 cos a = 3. (Gia sir cac bilu thiic da eho booktoan.com d6u cd nghia). 199
6.29. Cho tana + cota = m, hay tinh theo m 2
2
I
a) tan a + cot a ;
3
I
b) |tan a - cota\;
6.30. Cho s i n a + c o s a = m, hay tinh theo m a) sinacosa; 3
b) s i n a - eosal; 3
d) sin a + cos a.
c) sin a+ cos a ; 6.31. Chiing minh ring : , 1-cosa /l + eosa + xh a) J T 1 + cosa V 1 - cosa b)
1-1-cosa 1 - cosa
1-cosa V 1 + cosa
sina 2cosa sina
(Gia sit cac bieu thiic da cho d^u ed nghia).
§3. GIA TRI Ll/(?NG GlAC CUA CAC GOC (CUNG) CO L I £ N Q U A N D A C BifiT 6.32. Don gian hiiu thiic : a) cos a - — \ + s i n ( a - 7:) ;
b) eos(7i - a ) -i- sin a + —\ ; 7: c) cos — - a I -I- sin n l^a]-sin[^ — - a I - cos
+
aj;
f^Tt 1 . f37: ( In] . ( IK d) cos 1 ' ^ ~ < ^ I ~ ^ ^ " ^ ^ — ^ \ + ^os a - - — - sin a - —-
e) cos ~ - a \ + cosin - a) + cos -z— a \ + COS(2T: - a) ;
booktoan.com
200
3
c) tan a + cot a.
f) sin — - a
- COS — — a
g) cos(57: + a)-
- 3 s i n ( a - 57t) - 2 s i n a - e o s a ;
2sm — — a I - sinl — - -i- a
6.33. Chiing minh rang vdi mpi a ta cd : 57: \ . {3n a) sin —- + a = - s i n — — a 4 / \4 b) cos a - —
^
(
=^ -cos \^ + ^ j ;
2n]
]
(4n
c) cos Q^ - "T ~ '^^^ ~r ^ "^ " 6.34. Khdng su" dung may tinh va bang sd, hay tfnh : a) sin315° ; cos930° ; tan405° ; cos750° ; sinll40° ; b) eos630° - sinl470° - cotl 125° ; e) eos4455° - cos945° + tanl035° - cot(-1500°). 6.35. Tfnh 7:
27C
ST:
a)cos— -I- c o s — + ... + c o s — ; ,^.271
.2^
-2^
- 2 2n
. 2 5n
. 2 77:
b) sin - -i- sin -^ + sin — -h sin - ^ + sm -—- + sin -—; J o y y lo lo in 2 5n 2^ 211^ 2137: 22n c) cos — -I- COS -— -h COS — -I- cos — — -I- cos — — -I- COS -—; 3 71
6 27:
y
lo
18
y
9T:
d)eos-- -I- cos-— + ... -I- cos—-; , . 7:
.27:
.97:
e)sin— + sin-3- -i-... -i- sin-— 6.36. Gia sir tren dudng trdn lupng giac, di^m xac dinh bdi sd a nim trong gde phdn tu I, II, III, hay IV cua he toa dp vudng goc gin vdi dudng trdn dd (khOng nim tren cac true toa dp). Khi dd die'm xae dinh bdi eac sd : a + —; a + n ; cc - — ; -a z. ^ -a + n nim trong gde phin booktoan.com tu nao ? Diin vao bang sau :
; -a + ~; z.
201
Diem xac dinh bdi a n «+2
I
Nim trong goc phan tu II III
IV
II
a-f- 7: T:
"-2 -a n -a+ n 6.37. a) Tren dudng trdn dinh hudng tam O cho ba di^m M, A', P Chirng minh ring M, N la hai diem ddi xu^g nhau qua dudng thing OP khi va chi khi sdiOP, OM) + sdiOP, ON) = k2n ik e Z). b) Tren dUdng trdn lupng giac, xet cac die'm M, N, P xic dinh theo thii tu bdi eac sd a, p, y. Chiing minh ring M, A^ la hai diim ddi xiing nhau qua dudng thing OP khi va ehi \diia + p= 2y+ k2n ike Z). c) Tim di6u kien d^ hai di^m M, A^ tren ducfng trdn lupng giac xae djnh theo thd tu bdi eac sd a, p ddi xiing nhau qua dudng phan giac cua goc phin tu II (va IV) ciia he toa dp vudng gde gin vdi dudng trdn lupng giac. d) Hoi cae diem tren dudng trdn lupng giac xae dinh theo thii tu bdi cae sd — ; — ;•---; - — , cd phai la cac dinh ciia mdt hinh thang can hay khdng ? ^
^
VJ
1. jmi
6.38. Chiing minh ring, vdi mpi a, vdi mpi sd nguyen k, ta cd : sm\a
+ k—\ = ' (-1) sina
cos a + k— V 2
ne'u^ = 21
(-1) cosa
ne'u k = 21 + \ ;
(-1) cosa
ne'u/; = 21
(-1) ^ sina
neu k = 21 + I ;
tan a ne'u k = 21 (khi cac bieu thiic nay cd nghia). tan a + k—\ = \ 2J I - c o t a booktoan.com neu k = 21 +I
202
n K 6.39. Tfnh cos— vk sin — bang "phuong phap hinh hpc'' o
O
nhu sau : Xet tam giac vudng ABC vdi ^ 7:;; 7:,. 7: AC . n AB A = -;C = -thicos- = — ; s m - = — . Bing each xet dilm E tren canh AC sao cho AE = AB (h. 6.4), hay chiing minh ring :
n
V2W2
cos— =
-^
. Tc V2-V2
' sin— =
,.„„.. . , „ , , a sma 6.40. Chung minh cong thuc tan— = 2 1 + cosa ^ ^ 7:
(vdi 0 < a < —) bang "phuong phap hinh hpc'' nhu sau : n B = a Xet tam giac vuOng ABC vdi A = —, Bing each ve dudng phan giac BD cua gde B ., ^ r^ ^ ' t. , =' ^41) OC , . (h. 6.5), tu tinh chat -^-^ AB = -p^, BC hay suy ra rang : a ,,- , , 7: tan— = , sma 1-1- cos a Hay^ tmh tan —2 12 7:
6.41. Chiing minh cong thiie eos2a = 2eos a - 1 (vdi 0 < a < —) bing 'phuong phap hinh hpe'' nhu sau : Xet tam giac vudng ABC v6iA = —.B = a. Ke dudng trung true eua doan BC eat AB tai I. De thay : cos2a = -—- ; cosa = -—; /C £>C (h. 6.6); tur dd hay suy ra 2
Hinh 6.6
cos2a=2eos a- 1.
booktoan.com 203
§4. M O T n
TT
S 6 C O N G THL/C
71
n
LLfONG GlAC
IT:
6.42. a) Viet yy ^ T ~ T » j y ^ T - r
^^* dung edng thiic cpng, cong thiic
nhan ddi de tim eac gia tri lupng giac sin, cdsin, tang eiia goc — bing hai each khae nhau va ddi ehie'u cae ket qua tim tha'y. b) Tfnh sin, edsin, tang eua eae gde 75°, 105°, 165° (khdng diing may tinh bd tiii). 271
,
6.43. a) Tinh x = c o s — bang phuong phap hinh hpe" nhu sau : Xet tam giac can ABC vdi B = C = — . ke dudng phan giac BD cua tam giac dd. Tii tfnh eha't RC
DC
-— = -—- (h. 6.7) hay suy ra 4x^ + 2x - 1 = 0. BA DA b) TiJr dd tfnh cos-- ,sin-r-, tan — c) Tfnh sin, cosin, tang eua 18° d) Vi^t 6 = 36 - 30, tfnh sin, cdsin eua 6° Thii lai bing may tfnh bd tiii. 3 3 6.44. Cho c o s a ^ —; sina > 0 ; sinp=-, cosP< 0. Hay tinh cos2a, sin2a, cos2y5, sin2^, eos(a + p), sin(a - p). n a a a 6.45. a) Cho c o s a = 0,6 va 0 < a < —. Hay tfnh cos— ; sin— ; tan—. ^
Z^
Z^
z,
b) Cho s i n ^ ^ - v a ~< p< n Hay t f n h c o s ^ ; sin-y ; tan-^ 6.46. Cho cosa = m. a) Hay tfnh e o s 2 a ; sin^2a; tan^2a theo m (gia sir tan2a xac dinh). b) Hoi s i n 2 a ; tan2a cd xac dinh duy nha^t bdi m hay khdng ? 6.47. Cho sin a = m. Cung cau hoi nhu d bai 6.46. 204
booktoan.com
6.48. Cho c o s a - m. Hay tinh cos — ; sin — ; tan — theo m (gia su tan— xac dinh). Zd
^
Lt
^
a 6*.49. a) Tfnh sina, cosa theo tan— = r. , , T.~ ' , 1 - cosa 1 . . , a -i + 4 sin a theo tan— = / . b) Hay tinh sin a tan a 2 6.50. Gia sir cac bi^u thiic sau ed nghia, chiing minh ring : sina -I- sin2a ,^ 2 2sin2a - sin4a a) tan a = r—; b) tan a = ^ . ^ -.—-— 2sin2a -i- sin4a 1 -I- cosa -i- cosza 6.51. a) Chiing minh rang vdi mpi a, P, ta ed : 2
2
2
sin ia + p) = sin a+ sin p+ 2sinasinpcosia + p). b) Bie't cosa-I- cosp= m ; sina-i- sinp= n, hay tfnh cosia- p) theo m van. 2
2
c) Bie't cos a+ cos P = p. Hay tfnh c o s ( a - y^cos(a-i- P) theo p. 6.52. a) Chiing minh ring neu cos(a + p) = 0 thi sin(a -i- 2p) = sina. b) Chiing minh ring ne'u sin(2a-i-j^ = 3sin/?vaeosa?^ 0, cosia+p) ^ 0 thi tania+p) = 2tana. 6.53. Chiing minh a) 4cos 15°cos21°cos24° - c o s 12° - c o s l 8 ° = b) tan 30° + tan 40° + tan 50° + tan 60° = ^ , c)
1
1
\ + y/3
2
cos 20° ;
^ =2;
sin 18° sin 54° d) tan9° - tan27° - tan63° + tan81° - 4. 6.54. Chiing minh , sinx + siny ^ . X -I- y ,. . *5 , , = ^ ^ '^ a) < sin—-— VOI mpi x, y deu khdng am va :v + y < 2n. z. — * , , COSJt -I- COSy ^ x + y ,. . , ^ ~ ^ < COS——^ voi mpi X, y thoa man -n
^
6.55. Chiing minh sina + smp cosia + p) = tan(a + p) (khi cac bi^u thiie ed nghia). c o s a - sin;ffsin(a -i- p)
booktoan.com 205
6.56. Chung minh ring neu tam giac ABC thoa man dieu kien : , . . cosB + cosC a) sm>i = -^—r-^ sinfi + sinC , , sin^ cos6 + cosC b) ^ — - = ^ —sinS cosC + cos/l mpt tam giac can.
., thi tam giac ABC la tam giac vuOng ; ^ '^ ,, ., , thi tam giac ABC la mdt tam giac vuOng hOac •. ^ ^ .
6.57. Xet cac bidu thiic 5 = sina + sin2a -i- sin3a -i-... + s'mna, 7" = 1 -I- cosa + eos2a -i- eos3a -i-... + cosna in la mpt sd nguyen duong). Chiing minh : , ^ . a . na . in + l)a ,. _ . a na . in + l)a a) 5sin— = s m — s i n ; b) Tsm— = cos—sin-^——^-. 6.58. Chiing minh : , . 27: .47: .67: 1 TT a) s m — - -I- s m - — + s m — - = —cot—- ; 7 7 7 2 14 ,, 7t 371 57: 77: 97: 1 b) cos— + eos-— -I- c o s - ~ + '^o^TT + *^°^TT ^ T ' 27t
47:
67C
87:
IOTC
1
c) cos-— -1- cos-— -I- cos-— + cos-j— + cos—— = -— ; ,. . n . 2n . \0n n d) sin-— + sin-— -1-... + sm-—- = cot—11 11 11 22
BAI TAP 6 N TAP CHLTONG VI 6.59. Cho sina - cosa = m. Hay tfnh theo m b) Isina+cosal ;
a) s i n a c o s a ; 3
3
fi
c) sin a - c o s a ; 6.60. Tfnh
ft
d) sin a-i-cos a.
a) sin^l5° +sin^35° + sin^55° + sin^75° ; b)sin2f + s i n 2 ^ + s i n 2 ^ + s i n 2 ^ ; 0
0
0
8
c) cos^ - ^ + cos^ ^ + eos^ ^ + cos^ ^ + cos^ ^ + cos^ ^ 12 12 12 12 12 12 206
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6.61. Gia sir phuong trtnh bac hai ax + bx + c = 0, iac ^ 0) cd hai nghiem la t a n a va tanp. Chung minh ring a.sinia
+ p) + b.sinia +p)cosia + P) + c.cos ia + p) = c.
6.62. Chiing minh ring vdi mpi a ma sin2a -^ 0, ta cd 1 sin(cota) -f- sin(tana) = 2sin . ' |cos(cot2a). Vsin2ay 6.63. Chung minh cdng thiic c o s ( a - p) = cosacosp + sinasmp (vdi 0 < p< a< —) bing "phuong phdp hinh hoc'' nhu
sau
:
Xet
tam
giac
vudng
ABC
vdi
A = —; ABC = a ; E \a mpt didm tren AC sao cho Zi
ABE = p
Kc AH, EK vudng goc vdi BC (h.6.8) thi
>K 1 -' y rt, BK BH HK ^^ ^, de thay eos(a -P) =—- = - ^ + ^ T T - Tir dd suy ra DC tth tit. cdng thiie tren. 6.64. Chung minh ring cos —-• = - ^ 2 -i- ^ 2 -i- V2 + >/2 ^ Z*
Z
6.65. a) Chung minh eos — cos -— eos -— = - - bing each nhan ca hai ve' • 2^ v o^ i sin —
•, ^, ,
• u ^
2^
871
_
57:
T:
57:
b) Chung mmh rang cos - ^ -i- cos -— = 2 cos -— cos — ^ cos -—. ^- , . 2n 4n 871 ^ Tu do suy racos -— + eos -— + cos -— - 0 . ^ rrv . . i 2 27: 2 47: 2 87: 3 c) Tu b) suy ra rang cos -— -1- cos -— -1- cos — = — . '•'
7
^
Z
d) Tif b) va c) suy ra ring : 27: 47: , 47: 871 87: 2TC 3 cos — c o s — - h cos — c o s - ^ - H e o s - ^ c o s - ^ - - - .
booktoan.com 207
e) Tir a), b) va d) suy ra ring : /
2n^^ X - cos-— A - cos-— 27:
tCr dd ta cd 1 - c o s - ^
V
87:^
A - cos-— = X' ~^X
4n r 87: 1 - e o s - - X ~ eos - -
+ --
= -.
Suyra . n . 2n . 4n ^M ,sm-sm-^sm^ = —. . 57C . 77: . 87: v 3 . s m - ^ s m - ^ s m — = -g-. f) TiJr e) suy ra ring . 7: . 27: . 37: . 47: . 57: . 67: . 77: . 87: 9 si'^ 9 ^1*^-9-^1"-^ ^^"-9-^^"-9-^1" T ' ' " T ' ' " T ^ 256 (C/iH y. Ngudi ta chiing minh dupe ring khong the diing thude va compa de dung da giac diu chin canh npi tie'p trong mpt dudng trdn eho trudc). 6.66. Chiing minh ring O
0
cos iy - a) + sin (x - p) - 2cosiy - a ) s i n ( / - ;5)sin(a - P) = = cos ia -
p).
6.67. Tim gia tri be nha't ciia bieu thiic sin a + cos a . 6.68. Tim gia tri be nha't ciia bi^u thiic sin a + cos a.
Gidi THifiu MCyi s6 cAu HOI TRAC NGHISM KHACH QUAN Ddi vdi cdc bdi tic 6.69 den 6'.78, hay tim phuang dn trd ldi dung trong cdc phuang dn dd cho. 3n 6.69. sin-— bing : 47:
(A) cos-— ; 208
(B) c o sn- ;
booktoan.com
(C) 1 - c o s -
(D) - c o s - .
^ _„ . T: n . n 4n ,^ 0.70. sm—cos—r + sin—-cos-^ bang (A) 1 ;
(B) - i ;
iC)~;
(D) 0.
(C) V3 ;
(D) -V3.
(C) V3 ;
(D) -V3.
. n . 5n sin— -t- sin——
6.71.
9
1_ ^1
T:
JT:
cos— -I- COS — (A) -j^ ;
(B) — ^ ;
.57: . n Sin—-— sin — 6.72. ^ ^ bang 571 -7: eos-^-cos(A) ^
;
(B) -j^
•
6.73. Gia tri Idn nha't cua bieu thiic sin a + cos a la (A) 1 ;
( B) - ;
(C) - ;
(D) Khong phai ba gia tri tren.
6.74. Gia tri Idn nha't eua biiu thiic sin"* a + cos^ a la : (A) 2 ;
(B) 1 ;
(C) |
;
(D) Khong phai ba gia tri tren.
6.75. Gia tri be nha't eua bieu thiic sin a + cos a la : (A)-2;
(B)-l;
(C) - | ;
6.76. Gia tri Idn nha't cua bieu thu'c sin (A) 2 ;
a + cos
(B) i ;
(D) 1. a la :
(C) I ; 4
6.77. Gia tri nho nha^t cua bieu thiic
(D) ^
6
3 tan a la : cos a
(A) 4 ; 14-BTDS10,NC-A
(B) - 3 ;
(C) 1 ; booktoan.com
(D) 2. 209
6.78. Vdi mpi a, bieu thiic 97: n\ { 2n cosa + cos| a + —\ + cos a + — | + ... + cos a + V
nhan gia tri bing (A) 10 ;
(B) - 1 0 ;
(C) 0 ;
(D) Khdng phai ba gia tri tren.
C. DAP SO - HUONG DAN - LOI GIAI 6.1.
a) Sai : (OM, Ov) - a thi cd vo sd sd nguyen ^ de a+ k2n < 0. b) Sai : (Ow, Ov) = a thi (Ov, Ou) = -a + k2n, do dd ed vd sd sd nguyen ^ d e -a + k2n > 0. c) Sai : Vdi (Ow, Ov) - y va lay Ou' = Ov, Ov' = Ou thi (OM', Ov) = iOv,Ou) = -^nhungiidv
= ^u
d) Dung : ——- = 2n- — \ 6
= u^' = -2n - — ; uOv = — = u Ov
6
6
6
6
e) Dung : Vi hai gde lupng giac dd cd sd do dang a + k2n va a + 12K ik,i e Z), 0 < a < 2 7 : . %
7:
•"—~'
"—~'
f) Sai : vi (Ow, Ov) =— ; (Ov, Ou) = -— ed uOv - M'OV' nhung Z
n
6.2. a) 135° ;
b) 120° ;
O . a\) -^ ;
n
c) 330° ; d) ^ (77,1429)° ^ 77°8'34";
e) 2,3 ^ 131°46'49" ;. ^ 1
(
f) 4,2 ^ 240°38'32"
b, ), -57:;
c ),-27: ;
d ) -,.
5n
6.4. Gpi A, B la hai didm tiep xiie cua day curoa theo thii tu vdi dudng trdn tam / va tam / iA, B nim cung phfa ddi vdi dudng thing IJ). Ta co
booktoan.com 210
M-BTDSIO.NC-B
COS BJI =
= ^ — (r ^ 1 la ban kfnh cua dudng trdn tam /, ^ 2 ^ ^ d R ^ 5dm la ban kfnh ciia dudng trdn tam J, d = IJ ^ 8dm la khoang each -—n giua hai tam). Vay BJI = a = — De tha'y chieu dai day curoa bing : 2[R{K
- a) + ra + dsina~\ = 2( ~ ^ + 4^/3 j « 36,89 (dm).
6.5. Cac tia sang mat trdi ehie'u song song xudng mat dat: d Xy-en (kf hieu la S) ehie'u thing goc vdi mat dat, d A-le'ch-xang-dri (kf hieu la A) tao vdi phuong thing dung mpt gde (7,1)° nen sd do eung trdn AS la (7,1)° Gpi R (km) la ban kfnh ciia Trai Da't, thi do dp dai cung trdn AS bing 800km, 800 800.180 ^^^^^, ^ suy ra duoc R = ^ — - ^ 6456 ikm). 7:
7: X 7 , 1
^ ,
^
^
^^ ^ - - , ' . . ^ 4000000 ^ , , . , 6.6. Trong mot giay, banh xe quay duoc TTTFTm— ~ ^'4 (vong). 60.60.55.7:
6.7. a) Dien tfeh hinh quat trdn vdi ban kfnh R va goc d tam a la 2
S = ^ a = I R'^a. Tii do S = R^^a = 2. 2n 2 h) Chu vi hinh quat trdn ndi tren la C = 2R + Ra. Hai sd duong 2R va Ra cd tdng khdng d6i nen tich 2R.Ra - 45 dat gia tri Idn nhit khi va chi khi
2R = Rao
a=2.
c) Hai sd duong 2R va Ra cd tfeh 2R.Ra - 45 khdng ddi, nen tdng 2R + Ra=C dat gia tri nhd nha't khi va ehi khi 2R = Ra<=> a=2. 6.8. Dd dai eung kinh tuye'n dd la
^
'
'
^ 1558 (km).
loU 6.9. a) 420° ;
b) -612° ;
e) 390° ;
d) -1,72 ^ - 98°32'55"
6.10. a) « 0,349 ; b) « - 2,513 ; e) « 34,959 ; d) « 0,055. 6.11. Cac gde lupng giae (OM, OV) cd sd do la-- + ^27c = (lO^ + \)—.
k & Z.
3ln vay trong cac sd do da cho chi ed sd ——
booktoan.com 211
6.12. Cac sd a can tim theo thii tu la — \~ ; — ; a == 1,88971 = 5,934. 4 3 ' 6 ' 6.13. Cac sd a° cin tim theo thii tu la : 35° ; 28° ; 108° ; i20nf
(~ 62°49'55").
6.14. a) Ne'u mpt goc lupng giac (OM, OV) C6 sd do a, - T: < a < TT, thi mpi goc lupng giac (OM, OV) khae cd sd do a + k2n ik e Z \ { 0 | ) , nhung de thSiy a + ^27: 6 (-7: ; T:], vdi k nguyen khae khdng, vay goc lupng giac dd la duy nha't. Khi hai tia Ow, Ov dd'i nhau thi mdt gde lupng giac (Ow, Ov) ed sd do la TI va T: cung la sd do radian ciia gde bet uOv. Khi OM, OV khdng ddi nhau thi sd do gde hinh hpc MOV la /?, 0 < y?<7r va sd (Ow, Ov) \a p+ k2n hoac -P+k2n(k&
Z)tu-cla: sd(Ow, O v ) - a + ^ 2 7 : ; | a | - A
b) Sd do gde hinh hpc MOV can tim theo thii tu la c
c
'^
' T ' T ' T ' ^ 1,336 (do 2003 s. 319.27: - 1,336 va -TI < -1,336 < TC) ; • 140°; 125°; 145°; 157° 6.15. a) Viet a = a^ -I- k^^2n , - n< a^
Z)va
P^ Po+ lo^n , - n < p^ < n , il^ G Z), ta ed \a^\ la sd do ciiaMOv, \p^\ la sd do cua M'OV' Hai gde hinh hpc bing nhau khi va chi khi l«oH lAol ^ A = «o hoac a^ = -/?„ ^P~
a = k2n hoac P+ a = k2n, ik e Z).
b) Cap gde hinh hpc ung vdi cap gde lupng giae ^. .,, I3n . llTi ,. ^ , ^ ri37: II7: , ^ • Co so do —7- va —r- la bang nhau —-—h —-- = 47: . 6
6
\. 6
Cd sd do —— va ——- la bing nhau 212
booktoan.com
137:
0
J
UT:
-47:
.
. ^ , . . 17T: . va • Co so do 4 „,
..
73I7:
1571 ,, ^ . . fl77t la bang nhau 4 ^ ^ 4 .-117:,.^.
^
f73l7r
f 1571^ ^ ] —-— ^ 87: . V 4 ^ ) -UT:
^^ ^
. . ^ , ., , 20037: . -121171 ,, , . . • Co so do — - — va — - — la khong bang nhau 0
8
^^ 2003-M211 (do =
3214 , ^ . . . 2003-1211 792 ^^ —-— khong nguyen va = - - - = 99
khong chan). KTZ
TT
6.16. • N triing vdi M khi va chi khi cd sd nguyen / de —— = —+ /27: hay 798 6 k=\33i\
+ 12/).
Do i e N nen / e N. • N ddi xung vdi M qua tam ciia dudng trdn khi va chi khi cd sd nguyen / de ^^~+i2l+ 798 6
i)n<^k=\33il+
ni).Doke
N nen / € N.
6.17. Cach I. Diing hinh ve, de dang suy ra cac ke't qua sau ^ 137: • PN = PM o sd AP = —— + kn ik G Z) (cd hai diem P nhu the ling vdi Z^
k chin va k le). ^ 77: • NP = NM <^ sd AP = -- + k2n ik e Z). 6
• MP = MN <^sd AP = —— + k2n (/: e Z). Cach 2. Vdi ba diem phan biet M, N, P tren dudng trdn dinh hudng tam O gde A, de tha'y PM = PN khi va chi khi POM = PON nen theo bai tap 6.15 va do M khae A', ta ed sd iOP, OM) + sd (OF, ON) = k2n ik e Z), tiic la sd iOA, OM) - sd (OA, OP) + sd (OA, ON) - sd (OA, OP) = k2n ik G Z).
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^ 1 ^ r\ Vay PM = PN<^ sd AP = -(sdAM +sdAN) + knik e Z). Ttr dd suy ra cac ke't qua d each 1. , n n 6.18. • Cac diem tren dudng trdn lupng giac xae dinh bdi cae so— + k — ,ik eZ) la bdn diem cua hinh vudng ndi tie'p dudng trdn dd, ed hai canh song song vdi OA (O la tam, A la giao ciia dudng trdn vdi true hoanh (la gde ciia dudng trdn lupng giac)), (chi cin lay k^O, 1,2, 3). • Cae diem tren dudng trdn lupng giae xac dinh bdi cae s6k — .{ke Z), la eae dinh cua luc giae d^u npi tie'p dudng trdn dd, trong dd mpt dinh la gde A ciia dudng trdn lupng giac (chi cin lay k = 0, 1, 2, 3, 4, 5). 2n • Cac diem tren dudng trdn lupng giac xae dinh bdi cac so k-—. ik E Z), la cac dinh ciia ngii giac deu npi tie'p dudng trdn dd, trong dd mot dinh la goc A cua dudng trdn lupng giae (chi cin liy k = 0, 1, 2, 3, 4). 6.19.
120° -30° -225°
sin
cosin
tang
^/5 2
I ~2
-S
1 ~2
2
3
4i 2
750°
1 2
510°
1 2
2 2 2
booktoan.com 214
-1
Ghi chu
(-225 = -360+135)
3
(750 = 720 -1- 30)
3
(510 = 360+150)
sin
cosin
tang
4i
1
STT 4 771 2
2
2
-1
0
khong xac dinh
2
1 2
-^/3
57: T: - r - ^ 27: - — 3 3
2
1 2
-S
IOT: , 2T: — T - = - 4 T : + —3 3
2
1 2
- ^
——- = on - — 3 3
5T:
3 -IOTI 3 177U
3
Ghi ehii
In - ^
%
— - = 47: - — 2 2
1771
.
n
6.20. Diem xac dinh bdi a nim d gde phin tu II thi diem xae dinh bdi a - T: nim d gde phan tu IV.
a + — nim d gde phin tu III.
K
37:
— - a nim d goc phin tu IV.
- • - > , , -
TT
— a nam o goc phan tu II. 6.21. Kf hieu M la diem thupc dudng trdn lupng giac xac dinh bdi sd a thi :
sin
Dau cdsin
tang
3n n< a< — =>Me (III)
-
-
+
^ < « < ^ ^ M G ( I V )
-
+
-
^M
-
+
-
27: < a < 2 , 5 7 : = > M G (I) IO7: , - ,„_.
+
+
+
(III)
-
-
+
57:
.TTX
+
-
-
37:< a <
G (IV)
-r- =>MG 11^
,-
— < a< - — = > M e (II)
(Cac ki hieu (I), (II), (III), (IV) theo thii tu chi cae goc phin tu I, II, III, IV)
booktoan.com 215
6.22. M ed toa dp (x ; y) ^ (0 ; 0), dat sd (OJC, OM) = a thi x cosa =
2
2
. ; sm a =
y • vay
x^+y"
x +y cosa
M
sina
cota
tana
M(3 ; -4)
3 5
4 5
4
3 4
; -3)
4 5
3 5
3 4
4 3
4
3 5
3 4
4 3
4i
4i
2
2
-1
-1
M(4
M(-12;-9)
M ( - l ; 1)
6.23. A = - - ; 5 = 0. }^2n
6.24. • Cae diem tren dudng trdn lupng giac xae dinh bdi cac sd —— ik e Z) la cac dinh cua ngii giac deu npi tie'p dudng trdn dd ma mdt dinh la A(l ; 0). Tir dd quan sat hinh ta thay : k2n sin—— ik G Z) cd nam gia tri phan biet, k2n cos—— ik e Z) ed ba gia tri phan biet, ^27:
tan—— ik e Z) ed nam gia tri phan biet. ., kn • Cac diem tren dudng trdn lupng giac xac dinh bdi cac sd -— ik e Z) la cac dinh cua mpt luc giac d^u npi tie'p dudng trdn dd ma mpt dinh la A(l ; 0). Tir dd quan sat hinh ta thay : tan-— ik e Z) cd ba gia tri phan biet (cu the la 0 ; v3 ;-%/3 ).
booktoan.com 216
6.25. sinI0° ^ 0,174 ; e o s | « 0,940 ; t a n i ^ = « 0,364 ; cot(l,35) ^ 0,224. 5 / 25 12 6.26. a) c o s a = - p - . s i n a < 0 nen s i n a ^ - J l -7777 = -—r- d o d d t a n a = IJ
V
13
169
12
.
5
eota = 12 b) s i n a ^ - . e o s a < Onen eosa = - . 1 - ^r^ = -^. Ttrdd suy r a t a n a - — . 5 V 25 5 ^ 3 cota = —7. 4 15 _ . I I 8 . ^, . -—- = - — , tu do s i n a = c) t a n a ^ - ^ . c o s a < 0 nen c o s a ^ - | ZZ3 1/ 1+ 64
15
;
1/
cota = — . 1
d) cota = - 3 , sina < 0 nen sina = -
1+ 9
3 , tiJf dd cosa = VlO '
tana = — . 3 ,-_ 2sina + 3cosa 2tana + 3 9... 6.27, • ---. ^ = - khi tana - 3. 4sina-5eosa 4tana-5 7 3sina - 2 e o s a
3tana - 2
5sin a + 4eos a
cos a(5tan a + 4)
3 tan a - 2 /, 7 a)1 + tan 5tan a + 4 2
sin a
• 2
6.28. a) tan a - sin a cot^a - cos^a
70 khi t a n a - 3. 139 1 cos a
sin aten a
^^^2^
eos^aeot^a ^sin a
b)
-1
= tan a
J
sina + cosa - c o s a ( t a n a + l) / 2 .\/ x ^ = ^-r ^ = tan^a + 1 (tana + l) cos a cos a 2
3
= 1 + tana + booktoan.com tan a + tan a. 217
c) yjs'm a ( l + c o t a ) + c o s ' a ( l + t a n a )
= 4.sin a + s i n a c o s a •
2
•
2
+ eos a + c o s a s m a
-J(sina + c o s a ) ' = s m a + cosa . 2
2
2
2
2
d) sin a tan a + 4sin a - t a n a + 3 cos a = - t a n acos"'a + 4 s i n ' ' a + 3 c o s a 2
2
= 3(sin a + eos a) = 3. 6.29. Cho tana + cota = m, ta cd : 2
2
2
2
a) tan a + c o t a = (tana + cota) - 2 t a n a e o t a - m 2
2
-2.
2
2
= 4m^ (dey cotal Vay >2, b) |tana (tana - cota - cota) ^ 4tan a ring,dotana.cota^ + cot a - 2 t a n a cl,nen o t a Itana = m + -4. tiJt dd m^ > 4). 3
3
3
^
c) tan a + cot a = (tana + cota) - 3 t a n a c o t a ( t a n a + c o t a ) = m - 3m. 6.30. Cho sina + cosa = m, ta cd : m^-l
a) s i n a c o s a = — (sina + c o s a ) - 1 2
2
9
b) ( s i n a - cosa) = 1 - 2 s i n a c o s a = 1- (m - 1) = 2 - m ,
y
tir dd [sina - cosa| ^ V2 - m (lap luan nay cung ehung td ring, n^u s i n a + c o s a = m thi 2 - m >0, tiic la ta ludn cd jsina + eosal < V2 ; edn cd th^ suy ra bat ding thdc nay tiJf nhieu lap luan khae). 3
3
3
c) sin a + cos a = (sina + cosa) - 3 sina cosa (sina + cosa) = m -3 •J
1
m 3 -m")
^m'-i'
. 6
m= 2„.,3
^ . 2
r
-Sm" + 6m^ + 1 4
. 2
d) sin a + c o s a = ( s i n " a + c o s a ) - 3 s i n a cos a (sin a + cos a) wj-" -
= 1-3 218
booktoan.com
, ,,
,
ll-cosa
6 . 3 1 . a) 4[-
h + cosa _ | ( l - e o s a ) ^ + ''
l + cosa • "V 1 - c o s a
— *l
\|
1 - cosa + 1 + cosa
(l + cosa)'
^
+
sin^a
\
sin^a
. (Chii y ring Icosal < 1). sina
sina
-¥
(l + cosa)
\\ - cosa Vl + cosa
11 + c o s a V1 - eos a
\
sin a
- cosaj sin a
2 cos a 1 + cos a - 1 + cosa ' |sina| |sina| 6.32. a) 0 ; d) - 2 s i n a ; 6.33. a) sin
(5n
—T-^OL
c) 2 s i n a ; f)0;
b)0; e)0;
g) 2cosa. 37t
o 37: = sin 271 — Z - + a = sin 4
371
+ a h^ - sin
- a
^ 7:^ 27:^ ( 2n b)eos a - — - -COS a — r - + T: 1 = -cos a + 3 V
/ 47:"^ ( 47: ^ ^ ( 2n \ c)cos a — r - ^ cos a + —— 27: = eos a + 6.34. a) Dap sd theo thii tu la
_V2
_V3
2 ' «
^ --
-
V
!
2
1;
'
2
2
a-.4
•
7:
2T:
8T:
„
,
.
,
6.35. a)cos— + cos-— + ... + cos-— = 0, do cos(7: - a) = - cosa. b) Do sin— = sin
^n
n''
71
- 2 ^
• 2 "^
= cos— nen sin — + sin -r = \. 6 3 6
\ ^ .77: . (n n n . 2 '7^ • 2 ^ 1 D o s m - = s m - - - . = cos— nen sin -— + sm — = I ^ . 57: . 'n Do sm-— ^ sm lo
2n\
27:
,
. 2 2?:
. 2 5?:
,
._2 . 2 ^ . 2 ^ • 2 ^ . 2 27: . z ST: -^'v . z 771 "I. .2 vay sin — + sin — + sin — + sin -— + sin -7^ + sin 18 -r^= 3.
booktoan.com
219
c) Do eos
^ n n'^ T: 2 71 2 57: -^ = 1 • = eos 2 ^ 3 = - s m —i , nencos" —+ cos 3 D 6J
^57:^
71 T: UT: . n 2^ 2 11^ 1 Do cos—— - eos s i n — . n e n cos — + cos —^ = 1 • = 18 2^9J V . 27: „ 2137: 2 27: , ^ 13K 'n 2n] = - s i n - — , nen cos *>—- + eos -7^ =^ 1Do cos—— = cos 9 18 9 18 9 J
2^7: 2^ 2^1^ 21371: 2 2?: jn v a y COS — + cos - ^ + eos — + cos --— + cos —— + cos - ^ = 3 . 3 6 y lo lo y 27: ' n^ 7: 77: 671 = -cos— ; cos-— = - c o s - ^ ; d) D o COS-— = c o s 87: 371 97: 47: , . cos-z- = — cos*r- ; e o s - ^ = — cos-p- ; cos 7: — —1 nen 71 27: 97: , cos— + cos-3- + ... + COS-— = - 1 . e) Tuong tu dd'i vdi sin, nhung 0 day sin?: - 0, ta cd : . 7: . 27: . 971 ^ sin— + s i n - ^ + ... + sin-— = 0 . (Chii y : Ta cung ed th^ xet thap giac d6u ed cac dinh la A^ la eac di^m kn tren dudng trdn lupng giae, xac dinh bdi cac sd -— (k = I ; 2 ; 3 ; 4 ;... ; 9 ; 10) va nhan xet ringOA^ + OA2 + ... + OAy^ ^ 0 ) .
6.36. Diem xac dinh bdi a
I
II
III
IV
a+2
II
III
IV
I
a+ n
III
IV
I
II
n a-2
IV
I
II
III
-a
IV
III
II
I
n -a+2
I
IV
III
II
II
I
IV
III
-a+ n 220
Nam trong goc phan tu
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6.37. a) Theo md ta cua cung lupng giac, hai diem M, N tren dudng trdn dinh hudng tam 0 la hai diem ddi xung qua dudng thing OP iP thupc dudng trdn dd) khi va chi khi sdPM+sdPN
=k2nikG
Z).
b) Tir eau a) ne'u M, A', P thupc dudng trdn lupng giae xac dinh theo thir tu bdi cac sd a, p, y thi M, N la hai diem ddi xung qua dudng thing OP khi va chi khi a- y+ p-
y= k2n tiie la a + ^ ^ 2 / + k2n, (^ G Z).
e) Coi P xae dinh bdi sd — thi hai diem M, N xac dinh theo thd tu bdi a, p la hai diem ddi xung nhau qua OP (dudng phan giac ciia gde phan tu II va IV) khi va ehi khi a + P=^
+k2n.
d) Coi eac di^m Aj, A2, A3, A4 tren dudng trdn lupng giac xac dinh , •>. n n 5n \3n _ ,•,.,, • • . ^ . ^ ,v ,v , theo thu tu boi —; — ;-— ; ——. Ta phai ehung minh A1A2A3A4 la hinh 4 2 6 12 thang can. Cdch 1. Hai cap diem A] va A4 ; A2 va A3 ddi xung nhau qua cung mpt ,7: 137: 7: 57: 47: duong thang ^^ ~^^ ~^ = ^"^-^ ^ ^ n n n Cdch 2. Gde hinh hoe A I O A T ed so do — - - - = - - va gde hinh hoc \ A 2 4 4 ^ ^ . , - . 137: 57: 7: ^T'TTT' T"]^rT^ A3OA4 CO so do — 2~ ^ T ' ^^" A^'^2 ^ ^z^\ 6.38. • sin a + 21— 1 = sin(a+/7:) = ( - 1) sina ; n'\ / . ( sm a + [2l + \)~ = sm a + - + ln\ = (-1) sin a + ~ = (-1) cosa.
booktoan.com
I
V
221
COS
cc + 21- = cos(a + /7t) = ( - l ) ' c o s a ;
7:
= COS
eos a + ( 2 ' + 1 ) T
a + — + In = (-!)' COS a + -
= ( - l ) ' ( - sma) - (-l)'"^'sina. Tiidd tan a + 2/-
= tana ;
tan a + (2/ + 1 ) - = - c o t a . 6.39. Coi AB cd dp dai la 1 thi de thiy AE = AB ^ \,BE = CE = ^
AC = AE + EC=\+42 •BC=J\ + {l + ^f ^. ^,
7: AC
. n
I + V2
;
=^2(2 + 72).
V2W2
4^^
AB
g^uT) ^ ,^ ^ . AD 6.40. Ta CO AB _
, , AD (
DC BC AB
AC- AD BC
AC BC
AD AB AB BC
AC a tire la tan—(l + c o s a ) = sina, suy ra BC
a sma tan— 1 + eos a 2 1 ,.,. n . n ^-2 Voi a ^ — taduoe tan-— ^ —^ ^ r6 • 12 2 r 1 + V3^ 2 + V3
booktoan.com 222
-V^.
6.41. De tha'y BI = IC, AI IC
AI BI
nen cos2a = -— ^ —
AB-BI BI
'
^
AB BI
,
AB 2BM BC BI
1 = —T^-^TT,
,
1-
AB BM , , 2 , ma c o s a = —— = ^77-' nen c o s 2 a = 2cos a - 1. BC
BI
b) sin75° = eos—- ; 12
cos75°= sin—- ; 12
sinl05° = cos—- ;
eosl05° = - sin—- ;
12
tan75° =
T:
= 2 + 43.
tantanl05°=
12
TT
^^"12 sinl65° = sin— ;
eosl65°= - c o s — ;
LZ
tanl65° = - tan—•
I Z
1 ^
6.43. a) De tha'y BC ^ BD = AD, nen dat BC = a,AB^b
^
BC b-a ^ DC = — - suy ra DA BA ^ a
a b
Ta CO
1 0 27: l_2cos—
Tur (1) va (2) ta ed
-^-
. 2 7 :
...
= —. tuc la
^
thi c o s - ^ ^ ~
h
a
(1)
a b
... (2) ^ '
— = —•
2n
=2cos^ J
2C0S-r-
hay
4 e o s ^ — + 2 c o s ^ - 1 = 0, tiic la 4x^ + 2;c - 1 = 0. b) Giai phuong trinh (3), ta dupe x ^
hoac x
(3) •
^ . ^, 271 - I - V 5 r^n -.u^7: VS - 1 < 0 (loai) hoac cos-— - — Suy ra Tu do cos-— = 3
*T
booktoan.com
J
^r
223
^
1 + cos^
I^TTf
^_^
2n sin^ = ^
.
2 - ^
4
\\ - cos--
7:
c) sin 18" = sin— = sin
v2*5, f 1
eos 18° = '^'^^TTT ~ ^^^
K
sm — ; tan- = ^ = ^ 5 ^ 2V5 cos--
11
_^
-\W^)
^
11 + cos--
v2'5.
-hW^)1
t a n l 8 ° . ^ ^ ^ . j l - ^ cos 18° V 5 . n n n . n d) sin6° = sin(36°-30°) = sinf ^ - ^ = sin--cos-- - cos--sin — 5 6 5 6 1^5 6J n 1 V3 . 7: 1 = ^ s m - - - c o s - = - 6 5 - V 5 - K/5 + 1 (^0,1045). T: 71 cos6° = cos(36° - 30°) = cos ~ - — ^ 5 6 j
n n . n . n cos--cos— + sin—sin — 5 6 5 6
V3 7: 1 . 7: 1 ——COS-r + xsin-- = — 3(V5 + 1) + ^ 2 ( 5 - 7 5 ) ! ( « 0,9945). 2 5 2 5 8 6.44.
cos2a = — ; 7 cos2p = — ;
^
sin 2 a =
24 sin2^--—•
3f,
cos(a + j ^ = ~ 5 r sin(a-/?)---
3V7
77 "^^
^ ' !
booktoan.com 224
^ •' Gai y.
3 . r^ A • l^ 9~ 77 cosa = —, s m a > Onen s m a = \ i - T^ = —r i 4 V lo 4 sinp=-,
eos>9<0nen C0Sy5
^ ^- , a \\ + cosa 275 6.45. a) c o s - = J — ^ — = ^ - ;
25
. a sin-
5 '
1 - COS a _ 75
"~r
2
a 1 tan— - —. 2 2
b) cos^ = -^/l ~ 25 " ~ I '
'-I
'^^^T
t a n | = 3.
6.46. a)eos2a = 2eos a - \ = 2m - 1 ; 2
2
2
2
2
2
2
sin 2 a = 4sin a cos a = 4cos a ( l - eos a) = 4m ( 1 - m ) ; 2^ sin^2a 4m^(l - m^) tan 2 a = — =^ r-. eos^ 2 a (2m^ - 1)^ 7: ( n] i b) Khdng, chang ban cos— = cos -— = —, nhung [^ 3 J Z 3 . 271 73 sin—- = ——' sm 3 2 1
7^ 3 J 2
27:
27:
; tan—- - - 7 3 , tan —— = 73
2
6.47. a) cos2a = 1 - 2sin a = 1 - 2m ; . 2,
• 2
. 2
. 2
sin 2 a = 4 sin a c o s a= 4sin a ( l -sin a) = 4m (1 - m ) ; 2^ sin^2a 4m^(l-m^) tan 2 a = — z — = — • 7rT~ cos^2a ( lbooktoan.com -2m^)^ 15-BTDSlO.NC-A
225
27:1 b) Khdng, ching han sin— = sin — . 27: 73 . nhung sin-— = —-- sin \.'^
73 2
2
t a n ^ = -73, t a n f 2 . ^ U 7 3 . 'v • 3
, ^„ 2a 1 + cosa 1+m 6.48. cos — = = —-— 2 2 2 . 2« 1 - cosa 1- m sm -— = = —-— 2 2 2
Ta \- m tan" — 2 1+m
2t ^ ^ „ , . ^ . aa aa ^ a 20, z r ^ . , , a . . (gia su cos— ^ 0). 6.49. a) s m a = 2sm—cos— = 2tan—eos — = 2 2 2 2 1 + ^2 2 1 r ^ ., , a „. 2 <3r , 2 cosa = 2eos — - 1 = 1= ~ (gia su cos— ?t 0 ) . 2« 1 + r^ 2 1 + tan — 2 b) Khi sin a eos a ?^ 0, ta cd , . 1 1 - cosa 1 —: + + 4sma ^ — + 4sma. sm a tan a sm a a ;t 0 va f ?i 1, ta cd Vay khi t = tan— 1-cosa 1 , . r + 18r+ 1 —-. 1+ 4 sm a = — sma tana 2f(l +1 ) 6.50. a)
sina + sin2a
_
sina(l + 2cosa)
1 + cosa + cos2a
1 + cosa + 2cos^a - 1 =
b)
226
2sin2a-sin4a 2sin2a + sin4a
sinafl + 2cosa) 7 -r = t a n a . eosa(l + 2cosa)
2 s i n 2 a ( l - cos2a) 2 s i n 2 a ( l + cos2a)
booktoan.com
sin^a y = tan a . cos^a 15-BTDSlO.NC-B
2
2
6.51. a) sin ia + p) = {sina cosp + siny^cosa) = sin a e o s y5+sin jffcos a+2sinacosasiny5cosy9 9
9
9
9
= sin a ( l - sin ^ + sin pi\ - sin a) + 2sinacosasin;5cos^ 2
2
2
2
= sin a + s i n y f f - 2 s i n a sin ;ff+2sina cosa sin/?C0Sy(? 2
2
= sin a + s i n y^+2sinasinyS(cosacos>9-sinasin^^ 2
2
= sin a + sin j5 + 2sina sin^^ cos(a + p). 2
2
2
2
b) m +71 = (cosa + eos^^ + (sina + sin^^ 2
2
2
2
= COS a + s i n a + cos p +sin p + 2(eosacosP + sinasiny?) = 2 + 2cosia- P). „ . , . - m + n - 2 Do do cos(a - p) Z
c) c o s ( a - p)cosia + p) 1
.= -icos2a
\
0
1
+ cos2p) = - ( 2 c o s ^ a - 1 + 2cos^y? - 1) ^
Zt
= cos^a + cos^p - 1 =p-
I.
6.52. a) Ne'u cos(a +y^ ^ 0 thi sin(a + 2p) = sinacos2;ff+ sin2/fcosa = sina(l - 2sin p) + 2siny9cos/?cosa = sina + 2sinj^- sinasin>^+ cosacosy^ = sina + 2sin;ffeos(a + p) = sina. b) Ta ed sin(2a + ^ = 3sin^<=>2sinacosacos>5+ (2eos a- \)sinp= 3sin;ff • » c o s a s i n ( a + >^ = 2sin^.
(1)
Mat khae sin(2a + P) = 3sinp
<=> 2sinacosacosy5+ (1 - 2sin a)sin>9= Ssin^ff <=> sina cos(a + p) = sinp. (2) booktoan.com Tijr(l) va(2)suy r a c o t a t a n ( a + /?) = 2. Do dd tan(a + ;^ = 2tana. 227
6.53. a) 4cosl5°cos21°eos24'' - cosl2° - cosl8° = 2cosl5°(cos45° + cos3°) - 2cosl5°cos3° = 2cosi5°cos45° = cos60° + eos30° =
i + —
2
2 sin 90°
b) tan30° + tan40° + tan50° + tan60° -
cos 30°eos 60°
+
sin 90° cos 40°eos 50°
cos90° + eoslO° + cos90° + cos30°
4cos20°cosl0°
icoslO°cos30°
eoslO°eos30°
4-cos20° = ^ e o s 2 0 °
73
c)
3
1
1
sin54° - s i n l 8 °
2cos36°sinl8°
2cos36°
sinl8°
sin54°
sinl8°sin54°
sml8°sin54°
sm54°
2cos36° _ cos 36° d) tan9° - tan27° - tan63° + tan81° = tan9°+ tan81° -(tan27*' + tan63°) sin 27° eos27°
sin9°cos9°
sin 63° " +
cos63°
sin27°cos27°
2
2
sin 18°
sin 54°
= 2.2 = 4.
, _ . , smj: + smy . x +y x~ y ^ . x + y ^ , _ . , , , ^ 6.54. a) = sin—-—cos——-^ < sm——^. (Voi ehu y rang z
Zf
z
z
s i n ^ ^ > 0 do 0 < ^ ^ ^
Z
. , cosx + cosy x +y x-y b) ^ cos cos cos- ^ 2 228
<
T:
va c o s ^ ^ < 1) ^
^ x + y .,,,. , , , ^ < cos (Voi c h u y rang
> 0 do - - < <— va cos- „ booktoan.com 2 2 2 2
<1).
sina + sinygcos(a + ^ )
sina + - [ s i n ( a + 2p) - s i n a ]
cosa - sin ^ sin ( a + p\ ^ \ f^ f
^v ( 'yn\ 1 cosa + —[cos(a + 2p) - c o s a j
^
sin{a + 2y5) + sina 2sin(a + ^)cos>9 , . 7 \ ^ 7 \ ^ tan {a + P) • cos[a + 2P) + cosa 2cos(a + pjcosp ^
A A 6.56. a) Vl sinA = 2sin—tos— va 2 2 B+C B-C cosB + cosC _ 2 e o s - ^ — c o s ^ —
cos
sin5 + sinC ~ ^ . B + C B-C 2 sin—-— e o s — - — 2 2
sin
5_^1 T:
A
. A
sin — 2_ A cos— 2
cosB + cosC . ^ <=> 2cos^—= 1 <=> cosA = 0 nen de tha'y : sin A = —. „ sin S + sin C 2 • o A la goc vudng. b) Cdch 1 . A A . A B-C ' ^ " y '^^^ 2" _ " " " T ^ ^ ^ ^ — . B B . B C-A sm— cos— sin— c o s — - — 2 2 2 2
sin A _ eosfi + cosC sin 6 cosC + cos A A O
C~A
cos —cos
B
B-C
= cos — C O S — - —
<=> COS— + cosi A - — J = cosi B -—\
<=> COS
o
-f r
A- 2
+ COS—
= COS 5 - 2"
r z^-'
2
A =fi <=>
A +B
=C.
booktoan.com 229
Cdch 2 sin A sinfi
cosfi + cosC o sin Acos A - sinScosB = eosC(siii6 - sinA) cosC + cos A
— (sin2A - sin2S) = cosC(sinB - sinA) Z
B+A B —A C:> cos(A -fB) sin(A - 5 ) = 2 c o s C c o s — - — s i n — - — ^ . A-B A-B ^ . A-B A +B <=> - c o s C s i n — - — c o s — - — = -COSC s i n — - — c o s — - — ^ . A-B( A +B o cosCsin—z— c o s —
A~B , _ cos—-— I = U
cosC = 0 ^ . A . B . A-B ^ ^:>cosCsm—sm—sm—-— = 0 o 2 2 2 sin—-— = 0 6.57, a) Vdi jt = 1,2, 3
C vudng A = B.
n, ta cd
. . . a 1 smmsm—- = — 2 2
(2^-l)a COS-^^
—^
(2^ + l ) a COS-^
r—-—
nen a 3a 3a ^ . « 1 cos— - COS—— cos-+ S.sin—= — 2 2 2 2 2 (2«-l)a
+ cos-
5a 2 +... +
COS——
(2n + l ) a ^ - cosf a (2n + l)a cos— - cos 2 2
. na . in + \)a = sin——sm 2 2
b) Vdi ^ = 1, 2, 3, ..,, «, tacd (2k + l)a , . a 1 eos Arasm— = — sin-^^ —^ 2 2
booktoan.com 230
sin-^^
{2k-l)a -—^—
nen ^ . a . a 1 .5a .3a .3a . a L sin —= sin— + — sin—— sm— + 1 s i n — - s i n — + ... + 2 2 2 2 2 . (2n + l)a
+ sm
. (2n - l)a - sin
. (2/7 + l)a . a sin + sin— 2 2
2
na . (rt + l)a 2 2
= COS——sin
6.58, a) Ta cd . 2n . n \( n 3n s i n — s i n y = - eosy - c o s y . 4n . n 1 3n sin-—sin— = — c o s — 7 7 2
5n COS —
. 67: . 7: \( 5n sin—-Sin— = — cos—- - COST: 7 7 2 7 Tirdd .4% . (in\ . n if, 7: . 2n sin—- + sin—- + sin—- sin— = — 1 + cos— = cos 7 7 7 j 7 2 7 14 „
.
7:
_ .
7:
7:
Do sin— - 2sin—-cos—-. ta suy ra -^ 7 14 14 . 47: , . 67: 1 7: , 2n s i n - y + s i n — -i- sin— = ^ c o t — . b)Vdi^=l,2,3,4,5taed cos
{2k-\)n 11
. 7: _ 1 r . 2^7: . (2^-2);: sin-—— sin•^^" n = 2 L 11
nen n^u gpi B la ve' trai ciia dang thdc 0 cau b) thi . ^ ( . 4n . 2n\ f . IOT: . 871^ 1 ^ . 271 +^ s . n - - s m - J + . . . + ^ s m — - s m sin-— sinO ^^^^TT=2 „ . 7:
1 . IO7:
1.7:
TirddB=Z*
booktoan.com 231
c)V6ik=
1,2, 3, 4, 5 tacd
. ( 2 ^ + 1)TI . {2k-\)K 2^7: . TC 1 eos — s m - = - s m — ^ - s m ^ ^ ^
nen gpi C la v^ trai cua
ding thiic cau c) thi . 3Tr . T: . 971 .571 . 37C s m - - s m - + | s m - - s m - +... + sm7:-sm 11
^ - T: 1 Csm- = -
1 . T: = -2sm-. Tirdd C - - | . 2 d) Theo cau a) bai 6.57, gpi D la ve' trai cua dang thiic cau d) thi (d day =lO,a=^)
n
^
. n . lOn . n . IO7: n ^^"22 " ^ * " ^ ^*"2" " ^^"'22' " ^*^^22
Tiir do D = c o t ^ 22 6.59. Cho sin a - eos a = m ta cd , . in . .2 ,1 l-m^ a) s i n a c o s a = -— ( s m a - cosa) ~ 1 = — ^ b) (sina + cosa) = 1 + 2 s i n a c o s a = 1 + 1 - m = 2 - m' Tirdd sina + eos a\ = •
3
42^ m
3
3
c) sin a - c o s a = ( s i n a - c o s a ) - 3 s i n a c o s a ( s i n a - c o s a ) l-m^ m(3-m^) = m^ + 3 m = d) sin a + cos a = (sin a + cos a) - 3sin a cos a(sin a + cos a) = 1-3
1-m 2^
-3m'^ + 6m^ + 1
iChu y. Ciing dl dang suy ra eae ket qua nay tijf kd^t qua ciia bai tap 6.30 bang each dat a = n - a'). 232
booktoan.com
6.60. a) Vl sin75° = cosl5°, sin55° - eos35° nen sin^l5° + sin^35° + sin^55° + sin^75° = 2. b) Vi sin-— = sin . 2^
37:
n 37: .5?: ^K n^ = cos— ,= cos-— ; sm-— = sin 8^2
T:
T+2
. 2 37:
.
. 2 57:
-yln
nen sm — + sin -— + sin - - + sm —- = 2. c) Tuong tu UT: T: 57: . 5T: = -smcos 12 = cos 2" "^12 9T:
^n
\.Zf
J ^12, = -^'"I2
cos-— = eos
3n
.
3Tt
In 71 71 1 . n cos-— = eos 12 2^12 =-'""l2 nen ta cd : 2 3?: -^^
2 ^
2 "7?:
2 ^7:
cos ^2"^^°^ 'n'"^^^
2 9?:
12""^^*^^ l y "^
b
12" "^
2 1 1 ^ - .
I2" "
c
6.61. Tacd tana + tanjff= — . t a n a t a n j 5 = — a a • Ne'u eos ia + p) ^ 0 thi ve' trai eua dang thiie da cho la asin ia + p) + bsinia+p)
cosia + p) + ccos^ia + p)
= cos\a
+ p)[atan^ia + p) + btania + p) + c] 1 atan^ (a + p) + i?tan(a + p) + c l + tan^'(a + ;5)L
tan a + tan p Nhung tacd t a n ( a + i ^ = , 1 - tan a tan p
(*)
b c- a
(d^ y ring cos(a + p)^Ooc^a) ntn thay gia tri ciia tan(a +p) vao bi^u thiic (*), sau khi don gian ta dupe bieu thiic dd bang c. • Ne'u cos(a + /?) = 0 (*» tana tany?= 1 <=> a = c) thi sin^(a + p)=\, ?
'
^
nen
2
ve' trai eua dang thiic da eho bang asin ( a +p) = a = c.
booktoan.com
233
6.62. Dat u = —(tana + cota), v = —(tana - cota) thi M + v = tana, u ~ V = cota. Khi dd ta cd sin(tana) + sin(cota) = sin(« + v) + sin(H - v) - 2sin« eosv 1 f sin a
= 2 sin
cos a
cos a
+
sin a
ifsi sin a .cos 2 l+cosa
cos a sina
f • 2 2 \ 1 sin a -cos a 2 sin a cos a .cos 2 sin a cos a 1 = 2 sin (eot2a). sin 2 a .cos
= 2 sin
6.63. Ta cd cosia-p)
-
BK BH HK BH BA EJ + -^ iHKEJ la hinh chG nhat) BE BE BE BA BE BE BH BA EJ EA ^ . . ^ =^osacosP+sinasmP -BA-BE^-EA-BE
6.64, Ta cd cos— = —v2 ; '1 + cos— n cos— =
l-^-i^J^2
n cos- =
1 + cos^
ll + yll + Jl
1 r
cos-— = 32 6.65. a) Ta cd : . 2n 2n 4K 87: sin-— cos-— cos-— cos — 1 . 47:
47:
87:
1 . 87:
87:
= 2''''"9" '''''"9" ''^'T " 4''"T ''^''T 1 . I67: 1 . ^^ = -:rsin—— = —sin
2n\
1 . 27:
booktoan.com 234
r = ^
T,v, , ,
271
47:
87:
1
l u 00 :
cos—— cos—— c o s — = 9 9 9 8 u\T ' 27: 87: ^ 57: 71 5?: D) l a CO cos-— + cos-— = 2cos-— cos— = eos-— 47:^ 47: — = -COS
f = cos
7:
2?: 47: 87: ^ cos -5- + cos -— + cos -— = 0.
tirdd
c)Docos^:.2cos^|-l = 2eos2|i-l, 4J:
»
2 27C
,
cos-— - 2cos - - — 1 87c ^ 2 47: , c o s - ^ = 2cos - ^ - 1 , nen til b) suy ra 2 47: 2 8TI 3 2 27: cos - ^ + eos — + cos Y " 2 d) Vdi mpi sd A, B, C ta cd : 1 AB + BC + CA= - (A + B + C ) ^ - A ^ - B ^ - C ^ l nen 27C
47:
47:
87:
87:
27:
c o s - — C O S — + COS-^COS-— + COS-—COS-—
^
27:
47:
87:^
cos—— + COS—— + COS
9 J
[ -
2 2?: 2 4?: 2 8T: COS —— + COS —— + COS ——
9 J
2'2" 4' ^ 27:V„ 4n\(,, 87:^ e) Ta cd X - c o s - ^ X - cos-— X - c o s — =
2n 4n 87:^ X'- cos-— + c o s - ^ + cos-— x^ 27:
47:
47:
87:
+ 87:
27: ,,,
+ cos-—cos-—+ cos-—cos-—+cos-—cos-— IA 27: 4T: 871 ^ 3 3 ^ ! - c o s - ^ c o s - ^ c o s — = X - -X+ -•
booktoan.com
235
Tiidd
1 - cos
27:
47:^^
1-cos-^
V
87:
1 - e o s - ^ = - . tiie la
2 ^ o - 2 2 7 C 2 47:
3
2sm - . 2 s m - ^ . 2 s m ^ — = - , suy ra . 7t . 27: . 47: 73 sm-.sm-^.sm-^ = — ^. , , V , . , . 57: . 7;: . 87: 7 3 Dang thuc nay lai cho ta sin-— sin-— sm-— = - r f) Tir e) ta suy ra : .
7: . 27: . 37: . 47:
.
5T:
. 67: . 771 . 87:
sm-sm-^sm-^sm-^sm-^sm-^sm-^sm-^ ^^
73
. 7: . 27C
sin—sin—-- —— 3 3 256
6.66. Ta cd 2. ^ . 2. n^ l + eos2(y-a) l-eos2(y-yff) cos (^ - a ) + sin^(;' - P) = -^r^ + 7^—^ = 1 + - [ c o s 2 ( / - a ) - cos2(x - P)i = \ + sin(2/ - a - yff)sin(a - P). Tir dd 9
9
cos iy - a) + sin iy - p) - 2cos(/ - a ) s i n ( / - ;5)sin(a - p) = 1 + sin(2x -a
~ p)sinia
- p) - 2eos(/ - a)sm(;' - y9)sin(a - p)
= 1 + sin(a - y9)[sin(2/ - a - P) - 2cos(/ - a)sin(;' - p)'\ = 1 + sin(a - P)[sini2y - a - P)- s\ni2y - a - P)= 1 - sin2(a - p) = cos^ia - P).
sin(a - P)]
6.67. sin a + cos a = (sin a + cos^a)^ - 2sin^acos^a = 1 - —sm 2a. Zf
1
9
vay bieu thiie da eho lay gia tri be nhat la — khi sin 2 a = 1.
booktoan.com 236
6.68. sin^a + cos^a = (sin^a + cos^a)^ - 3sin^acos^a(sin^a + cos^a) •J
= l - 3 s i n acos^a = l---sin^2a. 4 1
9
vay bieu thdc da eho la'y gia tri nho nha't la — khi sin 2 a = 1. 6.69. Phuong an (B). 6.70. Phuong an (C). (De' y ring cos-— = -eos—). 6.71. Phuong an (C). 6.72. Phuong an (B). 6.73. Phuong an (A). (Dl y rang sin'^a < sin^a, eos'^a < cos^a). 6.74. Phuong an (B). (Di y rang sin''a < sin^a, cos^a < cos^a). 6.75. Phuong an (B). iDi y rang - s i n ^ a < sin'^a, - c o s ^ a < cos^a). 6.76. Phuong an (C). iDi y rang sin^^a < sin^a, cos^^a < cos^a). • 6.77. Phuong an (A). (De y rang — 3tan°a = 4(1 + tan^a)"* - 3tan^a chi cos a chiia nhflng luy thijfa bac chSn ciia tana vdi he sd khdng am nen nd dat gia tri nhd nha't khi tana = 0, |cosa| = 1). 6.78. Phuong an (C). (De' y rang cae di^m eiia dudng trdn lupng giae xae dinh , , . , , , n 2n 97: „ , ., , , ^ , boi cac sd a, a + — . a + —-. ..., a + -— la cac dinh cua mOt thap giac deu npi tiep dudng trdn dd hoac de y ring
r
57:'l
(
n]
f
6n]
^
cosa = -COS a + -— , cos a + : - I = -cosi a + -— \, ...).
booktoan.com 237
BAI TAP ON TAP CUOI NAM
A. DE BAI 1.
Cho
A = iO;4),B={xe C={A:G
E|A:^<32},
R| (x-2)^>3},D-[76;+oo).
Tim(AnB)^(Cr.D). 2.
Cho X la sd vd ti va a, b, c, d la cac sd hiin ti sao cho a J - be ^ 0. Chiing ,. ax + 6 ... ,, , , . . ^ la so vo ti. minh rang so ex + d
3. »
Cho menh de chiia bie'n Pin) : "Ne'u tong cac chU sd cua sd nguyen duong n chia he't cho 6 thi n chia he't cho 6". Pin) la menh de sai khi n bang (A)30 ;
4.
(B)33 ;
(C) 40 ;
(D)42.
a) Su dung may tinh bo tui de tinh 72006 , may tfnh cho ket qua la 44,78839135. Hay cho bie't dp chinh xac d ciia ke't qua nay. b) Khi vie't a a; 15,7 ± 0,3, ta hiiu sd diing a nam trong khoang nao ?
5.
Cho ham s6fix) - - j c + | ; c + 2 | - | x - 2 | . a) Hay vie't ham sd dudi dang ham sd bac nha't tren timg khoang va khdng chiia da'u gia tri tuydt ddi. iGai y. Xet ham sd tren mdi khoang (-co ; -2), [-2 ; 2) va [2 ; + ^)). b) Chiing minh rang y =fix) la ham sd le. c) Ve dd thi eiia ham sd y ^fix), lap bang bien thien va neu su bien thien cua nd tren mdi khoang k^ tren. d) Sir dung dd thi, hay tim cac khoang tren dd ham sd cd gia tri duong.
6.
Cho ham sdy = x - 4x + \. a) Khao sat va ve dd thi iP) ciia ham sd da cho.
booktoan.com 238
b) Gpi id) la dudng thing song song vdi dudng phan giac eiia gde p h ^ tu (/) va di qua diem A/(0 ; m). Xac dinh bieu thiic ciia ham sd ed dd thi id). c) Tim hoanh dp cac giao diim AvaB (ne'u cd) eiia id) va (P), va toa dp trung diem eua doan thang AB (khi A vaB phan biet). 7.
Giai va bien luan cac phuong trinh sau theo tham sd m : a) m^x - 3m^ = 9(x + m) ;
h) mix + 6) = x + 2m^ + 4 ;
c) |mx + x - l | - U + 3 | = ^ 0 ; ^ X+ a X- a e) a - X + a + X 8.
d) | m x + l | ^ | 2 x + m - l | ;
a (^ - ^^
Giai cac he phuong trinh sau : fO,lx - 0,3y = 0,7 [x - 3y - 7 ;
9.
|75x - 73y = ^ |6x + 72y - 75.
Cho he phuong trinh : {a + 4)x + ay = 2(o + 1) (I) 1
\ia + 2)x + 2ay = 1. a) Giai va bien luan he (I) theo tham sd a. b) Khi he (I) cd nghiem duy nha't (x ; y), hay tim he thirc giiia x va y khong phu thupc vao a. 10. Giai eae phuong trinh : 13 1 a ) 2- ^x ^ + x - 2-1 + 2x + 7
6 ;^-2_9
,,x+l
x+4
x-2
x-3
.
b)7TT + - ^ T 2 ^ 7 T 3 ^ 7 ^ = '11.
Tim ta't ca cae gia tri ciia tham sd a d^ phuong trinh : ia + 2)x- + 2ia + \)x + a-2 = 0. a) Cd hai nghiem khae nhau. b) Cd it nha't mdt nghiem. e) Cd hai nghiem bang nhau.
booktoan.com 239
12.
Dimg dd thi de bien luan sd nghiem ciia phuong trinh : x^ - 6x + 3 + m = 0.
13.
14.
2
Gia sir Xjva X2 la cac nghiem cua phuong trinh bac hai ax + bx + c = Q trong dd ac ^ 0. Hay bieu dien cae hiiu thu:c sau day qua cae he sd a,b,c: 7 7 2 2 a) X2X1 + x^xj ; b) xi - X2 ; c) x^ - X2. Giai eae he phuong tnnh sau : [x^ + y^ + X - 2y = 2 ;
[(2x - y)y = y ;
^ \5ix + y) + 2xy = -19 c) [3xy + x + y = -35. 15.
So sanh cac sd sau day : a) 72003 + 72004 va 72000 + 72007 ; b) 7/7 + 3 + 7n + 4 va 7n + 7 " + 7 (vdi rt > 0) ; c) yfa + \lb va y/a -c + yfb + c ,v6\ a> b>
16.
C!hiing minh cae ba't dang thiie sau : a)
b) -2
> 2
V?T2 17.
oO.
< X-
«' + i 2-
a) Chung minh rang dd'i vdi ba sd a, b, c tuy y, ta cd \a\ + \b\ + \c\>\a + b + c\. Dang thiic xay ra khi nao ? b) Ap dung. Tim gia tri nho nha^t eua bilu thiic fix) = |x + 2| + |x + l| + |2x - 5|.
18.
Cho ba sd duong a, b, c. Chiing minh rang : a) ac + - > 2yfab ; c
h) -^ + ~ > 2%b. ylb 7a
Trong mdi ba't ding thiicbooktoan.com tren, da'u bang xay ra khi nao ? 240
19. 20.
Tim gia tri nhd nha't eua ham sd gix) =x+
vdi x > 2 ; x-2 a) Chiing minh ring vdi mpi sd thue a, b, c, x, y, z ixyz ^ 0), ludn cd (ax + by + czf < ia^ + b^ + c\x^
+ y^ + z^).
Da'u dang thiic xay ra khi— = — = - . x y z b) Ap dung. Cho x^ + 2y^ + 3z^ = 6. Chung minh ring Ix + 2y + 3z| < 6. 21.
Giai cac he ba't phuong trinh 2x + 1 < X + 4 X + 1 < 2x - 3 a)
5-3x
ux , X + 20 b) -^ X + 1 < — - —
> x + 3;
x7x + 1 > 0. 22.
Giai va bien luan he ba't phuong trinh : f l + W1X > 0
| x - 2 < 0. 23.
Giai cac ba't phuong trinh : a)
24.
1-x <0; (2x - l)(x - 2)
b)^±l> ^-^ 2x + 1
Giai cac ba't phuong trinh : a) |x + ll + 3|x + 2| > x + 7 ;
25.
3x + 1
b)
-5 < 10 x+2 x-1
Xac dinh mien nghiem cua cac he bat phuong trinh sau ; y<2 2y - X > 2
x>-l a)
b)
x-y <1
X + 3y < 3.
X+y<2; 26.
4x + 3y > 12
Tim cae gia tri ciia m di mdi bieu thire sau day ludn duong : a)im^ + l)x^ + ( m - l ) x + 3 ; b) ( 7 2 - m)x^ + im -yl2)x
16-BTDSlO.NC-A
+ 2m + 3 42
booktoan.com 241
27.
Tim cae gia tri cua m di mdi bidu thiie sau day ludn am : a) - 4x^ + (4ffl + 7 2 )x - m^- 7 2 m + 1 ; b ) ( 5 m + l ) x ^ - ( 5 / ? i + l ) x + 4m + 3.
28.
Giai eae ba't phuong trinh sau : 3 ^8 +. 4x ... 2 4x +
29.
x'-2
X
4 +X
1
x'-4
>
(X + 2)'
_L 2x
Giai cac he bat phuong trinh sau : x^ - 1 4 x + 4 5 < 0 a) < x^ - l l x + 3 0 > 0 ;
30.
b)
-x-6<0
b)
x^ + x^ - X - 1 > 0.
Tim ta't ca cae gia tri eua tham sd m di cac b^t phuong trinh sau vd nghiem : a) 3x^ + mx + m + 2 < 0 ;
b) (3 - m)x^ - 2(2m - 5)x - 2m + 5 > 0.
31.
Tim cac gia tri ciia m di phuong trinh x - 2mx + m - 1 =^ 0 vd nghiem.
32.
Tim cac gia tri eiia m di phuong trinh im - l)x^ - (m - 5)x + m - 1 = 0, cd hai nghiem phan biet Idn hon - 1 . Giai cae phuong trinh :
33.
b) 4x'^ - 37x^ + 9 - 0 .
a)x'^-18x^ + 8 1 - 0 ; 34.
Giai cac bat phuong trinh : a) 7 - x ^ + 4x - 3 < X - 2 ;
35.
Giai cac ba't phuong trinh a) 2x^ + 72x^ - 8 < 20 ;
36.
b) 7 2 x + 5 > x + 1.
b) x^ - 3x - 7x^ - 3x + 5 > 1.
Mdt nghien cihi v^ tu6i ciia nhiing phu nO Mi sinh con l^n ddu cho ta sd lieu sau : Tudi cua me
Tdnsd
[15;19]
312448
[20 ; 24]
350905
[25 ; 29]
196365
[30 ; 34]
94874
[35 ; 39]
34408
booktoan.com 242
N = 989000 16-BTOSlO.NC-B
a) Da'u hieu la gi ? Don vi dieu tra la gi ? b) Tim tu6i trung binh cac ba me d Mi sinh con l^n ddu. c) Lap bang phan bd tdn sua't. d) Ve bidu do tdn sudt hinh quat. e) Ve bieu dd tdn sudt hinh cOt. 37.
Tim ta't ca eae mdu sd' lieu kieh thude 5 ed eae tinh ehdt sau : - Cac sd lieu trong mdu la cac sd nguyen duong. - Sd trung binh la 12, sd trung vi va mdt deu bang 8. - Bien dp (hieu giiia gia tri Idn nha't va gia tri be nhdt eiia mdu) bang 18.
38.
Chiing minh rang neu sinfa -P) = -sinp, thi tan(a -p) = 3 ' 3 + cos a
39.
(Dhiing minh ring, ne'u a + p+ y = 7: thi 2
2
2
cos a+cos p+cos 40.
1.
Tim gia tri nho nha't eiia bidu thiic . 1 -eosa 4. 1 + cos a
41.
y+2cosacospcosy=
2 + 3^^ /(gia - su' cos— ^^r ^ 0). „, 2CC 2 cos -
a) Vdi cac gia tri nao eiia a thi bidu thiic sau day cd nghia ? sina + sin 3a + sin 5a + sin 7 a cosa + cos 3a + cos 5a + cos 7 a b) Chiing minh rang vdi eae gia tri dd ciia a thi bidu thdc da cho bang tan4a.
42.
Chiing minh rang vdi mpi a, p, ^^ ta ed : cos^a + cos^p - cos^y - cos^(a + p + y) = 2eos(a + p)smip
+ / ) s i n ( / + a).
B. DAP SO - HUONG D A N - LOI GIAI 1.
A = (0;4);
B - ( - 4 7 2 ; 4 7 2 ). Vay A n B = ( 0 ; 4 )
C = (-oo ; 2 - 7 3 ) ^ ( 2 + 73 ; + o o ) ; D = [ 7 6 ; + c o ) .
booktoan.com
243
V a y C n £) = ( 2 + 73 ;+oo). Vay (A n B) u ( C n D) = iO;+oo). Chirng minh bang phan ehung. Gia sii
r = r la sd huu ti. Khi dd ex + d b-rd ax + b = rd + rex. Vay x(rc - a) = b - rd. Ne'u re - a 9^ 0 thi x = re -a la sd hiru ti, trai vdi gia thie't. Vay re = a do dd rd = b. Nhung khi do ad- be = red -red = 0. Dieu nay trai vdi gia thid^t. Gia tri n de Pin) sai khi tdng cac chii sd ciia n chia hd^t cho 6 nhung n khdng chia het cho 6. Chi ed duy nha't gia tri n ^ 33 thoa man dieu nay. vay cau tra ldi \aB). a)d =
10"^ L
b ) a G [15,4; 16] -x-4 a ) / ( x ) = \x
khi X < - 2 khi-2
- x + 4 khi X > 2. b) Tap xac dinh ciia ham sd la R. Vdi mpi x, ta ed : Cdch 1. (sir dung tinh chat |-a| - \a\) : f{-x) = -i-x) + I (-X) + 2| - \i-x) - 2 I = X + |x - 2| - jx + 2| = -(-X + |x + 2| - |x - 2|) =
-fix).
Cdch 2. (sir dung ket qua eau a) : - Neu X < - 2 thi - x > 2, nen/(-x) = - ( - x ) + 4 = - ( - x - 4) = -fix). - Ne'u - 2 < X < 2 thi - 2 < -X < 2, nen/(-x) = - x = -fix).
244
- Neu X > 2 thi -X < - 2 , nen/(-x) = -i-x)
-4 = -i-x + 4) = -fix).
Vay trong mpi trudng hpp ta deu cd fi-x) sd le.
= -fix), chiing td fix) la ham
booktoan.com
e ) D 6 thi (h. 1). Bang bidn thien X
-2
-00
+00
+00I
\
-2
—00
d) (-00 ; - 4 ) va (0 ; 4). a) Hpe sinh tu giai.
//,'„ft 7
b) Ham sd edn tim \ay =x + m. c) Hoanh dp giao didm ciia id) va (P) la nghiem eua phUOng trinh 2
X - 4 x + l - x + m, 2
hay
(1)
X - 5 x + l - m = 0.
Phuong trinh (1) ed biet thiie A = 25 - 4(1 - m) = 21 + 4m. Do dd, ne'u 21 + 4m > 0 thi nd ed hai nghiem 5 - 721 + 4m . 5 + 721 + 4m ^1 = 2 ^^-^2= 2 Dd cung la hoanh dp cac giao didm A va B cua id) va (P). Toa dd trung didm ciia doan thang AB (khi A = 21 + 4m > 0) la didm cd tpa dp (XQ ; yo), trong dd : XQ
-
Xj + X2 _ 5
2
yQ = XQ +
" 2 ' m=-+m.
a) Ta cd m^x - 3m^ - 9(x + m)
thi phuong trinh cd nghiem duy nha't x =
3m m-3
- Ne'u m = - 3 thi phuong trinh cd dang 0.x = 0, nghiem dung vdi mpi X e M. Tap nghiem 5 = R. - Neu m ^ 3 thi phuong tnnh booktoan.com ed dang 0.x = 36 (vo \x). Tap nghiem
S=0. 245
b) Bie'n doi phuong trinh ve dang (m - l)x = 2(m - l)(m - 2). Phuong trinh ed nghiem duy nha't x = 2(m - 2) khi m ^ 1 va nghiem diing vdi mpi X e R khi m = 1. c) jmx + x - 1| = |x + 3i
(1)
« • mx + x - l = x + 3 hoac mx + x - 1 = - x - 3. i) mx + X - 1 = X + 3 <=> mx = 4.
(2)
- Khi m = 0, (2) trd thanh 0.x = 4 nen phuong trinh vd nghiem 4 - Khi m ^ 0, (2) cd mot nghiem x = — m ii) mx + X - 1 = -X - 3 o (m + 2)x = - 2 .
(3)
- Khi m = - 2 ; (3) trd thanh 0.x = - 2 nen phuong trinh vo nghiem. -2 - Khi m^-2;
(3) cd mdt nghiem x =
-• m+2
Ket ludn.
Vdi m = 0, phuong trinh cd nghiem x = - 1 ; Vdi m = -2, phuong trinh cd nghiem x = - 2 ;
m m 2+ 2 4 Vdi m^O, m^ - 2 , phuong trinh cd nghiem x =^ — va x = d) Vdi m = 2, tap nghiem S = R. Vdi m = - 2 hoac m = - 1 , phuong trinh c6 nghiem x = 1 ; Vdi m^2,m^
- 2 , m^-\,
phuong trinh cd nghiem x = 1 v a x =
m+2
e) Didu kien ciia phuong tnnh : x ^ ± a. Ta dua phuong trinh vd dang 4ax = a.
(1)
• Neu a = 0 thi (1) cd dang OJC = 0, phuong trinh (I) nghiem diing vdi mpi X e R. Vay phuong trinh da cho nghiem diihg vdi mpi x G R
booktoan.com 246
• Ne'u a ^ 0 thi (1) cd nghiem x = —• Xet dieu kien x ^ ±a, ta ed — = ±a <=> a = ±—- vay khi a =^ 0, a ^ ±—thi x = — la nghiem cua phuong trinh da cho. Ket ludn : Vdi a = 0, tap nghiem ciia phuong trinh la 5 = R ; Vdi a = ~ hoac a = -—, tap nghiem ciia phuong trtnh la 5 = 0 ; Vdi a^O, a ;t ±—, tap nghiemS = — X G
8.
a) He cd v6 sd nghiem y =
x-7
b) Ta cd D - 7l0 + 673 ; D^ - 2 + 7l5 ; D^ = 5 - 6>^. He phuong trinh cd nghiem duy nhdt ix;y) =
^ 2 + 75
5-6>l2 ^
7io + 673' 7io + 673
a) He cd nghiem duy nha't (x ; y) = |
, ; a +6 '
ta +6
1 ne'u a ^ 0 va
a ^ -6. He v6 nghiem ne'u a = - 6 ; He cd v6 sd nghiem x ; y ) vdi y tuy y ne'u v^ a = 0. b) Khi a 5t 0 va a ^ -6, he ed nghiem duy nhdt ix;y) =
4a+ 3 ^ -i2a + 5) a +6 ' a +6
^ 4a+ 3 , 3-6x ^ ,, Do X = — nen a = -• Do do x-4 a +6
booktoan.com 247
y =
-(2a + 5) a +6
x +2
X-4
3-6x +6 x-4
x+2
vay khi he cd nghiem duy nha't (x ; y) thi y = — - — 10.
a) X = - 4. . ,_ , b) Ta cd
x +1 2 x~2 = 1r = 1 + x-1' x+2 x-1 x +2
6 x-3 = 1x+3 x+3
x+4 x-4
= 1+
nen phuong trinh da cho trd thanh : hay
5x-8 (x - l)(x - 4)
x-4
I x-1
2 X+2
3
x+3
+ ^
x-4
=0
5x + 12 (X + 2)(x + 3)
Tir do phuong trinh da eho tuong duong vdi he r(5x - 8)(x + 2)(x + 3) - (5x + 12)(x - l)(x - 4) (*)
[(x - l)(x + 2)(x + 3)(x - 4) ?t 0 Phuong trinh thii nhdt eua he (*) dupe bie'n ddi thanh phuong trtnh X + x — — =Ovacohai nghiem XJ = —
...,f
1 vax2=-
-,,f V
Vi hai nghiem nay thoa man dieu kien thii hai cua he (*) nen chiing la nghiem ciia phuong trinh da cho. 11.
a) De phuong trinh cd hai nghiem phan biet dieu kien edn va du la fa + 2 ?^ 0 A' = (a + 1)^ - (a^ - 4) > 0
<=>
a
^-2
[2a + 5 > 0
o a e ( - - ; - 2 ) u ( - 2 ; + oD).
booktoan.com 248
b) Xet cae trudng hpp sau : • (3 + 2 = 0 <^ a = -2 khi dd phuong trinh trd thanh - 2 x - 4 = 0 x--2.
• a + 2 ?t 0 <=> a ^ -2. De phuong trinh cd it nha't mpt nghiem, didu kien edn va du la : A' = (a + l)^-(a^ - 4) > 0 2a + 5 > 0 <^ a > - | . Vay a e [ - - ; + oo). c)a = --12. Xet ham sd y =/(x) = x^ - 6x + 3. Do thi ham sd la mpt parabol quay be 16m len tren (h.2) va dinh parabol la didm 7^(3 ; -6) , _. Do dd parabol ed phuong trinh 2
'
y = X - 6x + 3 va dudng thang cd phuong trinh y = -m : + Cd mdt didm ehung duy nhat khi m = 6 ; + Cd hai didm ehung phan biet khi m < 6 ; + KhOng cd didm ehung khi m> 6. Suy ra phuong trinh x - 6 x + 3 + m = 0 Hinh 2
+ Cd nghiem kep khi m = 6 ; + Cd hai nghiem phan biet khi m < 6 ; + V6 nghiem khi m > 6. 7
13.
7
^^
a ) X2X1 + X1X2 = XjX2(Xi + X2) =
I
— ^
a
b^ I b -^ 4ae
b) Ta ed |xj - X2I = ^jixi + X2) - 4x^X2 = J booktoan.com V 17-BTDSlO.NC-A
a 249
Suy ra : Ne'u Xy - Xi ^ 0 thi Xj -
XT
Ne'u Xj -
XT
XT <
0 thi XJ -
b^ - 4ae
-
= -,
Ib^ - 4ac
c) Xj - X-, = (.Vj - X2)(xj + XT). SU dung ket qua cau b) ; ^ ^ L^ '' > 0 thi X^ -
N e u Xj -
XT
Neu X, -
^ /A . V 2 XT < 0 thi X|
a)(l ;2)va
'
-
23 _ 44 17 * 17 )
b) Nghiem eua he la:
b
"• XT
=
-> b XT = -
-4ac
lb" ,I
lb
A ^ V
r
-
4ae
a~
•
rv7 \
2
; 0
( V7 1
) *>.
2
0 ,(-l;-3)va )
-;2
Gai y. TU phuong trinh thii hai suy ra y = 0 hoac y = 2x-1. e) Nghiem ciia he la (-3 ; 4) va (4 : - 3 ) . 15.
a) 72003 + 72004 > 72000 + 72007 ; b) yln + 3 + yfn + 4 > yfn + yJn + 1 (n > 0) ; e) Nhan thay f 7 a + 7^1 = a + b + 24ab Na -c + ylb + c]
= a + b + 2-^ia -c)ib
+ c);
Do (a - c)ib + e) = ab- cia - b) - c^ b> e> 0), nen 27(a - c)ib + e) < 24ab
Vi vay -Ja + 4b > ^la-c
+ ylb +
booktoan.com 250
17-BTOSlO.NC-B
16.
a)
^ '^
fa ^+2
>2 <:> a^ +2 + l> 2\la^ + 2 ^
i^a^ + 2 - 1)^ > 0.
Do a + 2 > 2 vdi mpi a ntn yja^ + .2 - 1 > 0. Vi vay bdt dang thiic cudi Cling dung. Suy ra dieu phai chiing minh. b) -^— < ^ « a +1 ^
2a^ < a^ + 1 (a^ _ 1)2 > 0 (dung),
(Da'u dang thiic xay ra khi a - 1). 17.
a) |a| + |^>| + |c| = i\a\ + \b\) + \c\ > |a + 6| + |c| > |a + ft + cj. Dang
thuc
a<0,b<0,e
xay
ra
khi -^ [ia + b)e>0,
tire a > 0, 6 > 0, c > 0 hoac
<0.
b)/(x) - |x + 2| + |x + 11 + |2x - 5| - |x + 2| + [x + 1| + |5 - 2x| >|x + 2 + x + 1 + 5 - 2 x | = 8. Dang thdc xay ra, chang ban tai x = 1. Vay gia tri nho nhdt ciia/(x) la 8. 18.
a) Vdi a > 0, & > 0, c > 0 ta ed ac + — > 2.\ac.— = c \ c
24ab.
2 b Dang thiie xay ra khi ac = — hay b = ac c ,\ a b ^ ab ^Arr b) -= + -==> 2, - p = - 2ylab.
sib
7a
\ylab
Dang thiic xay ra khi a = b. 19.
b) X + ^ ^
= X- 2+- ^
+ 2 > 2^(x - 2)—^
+ 2 ^ 4 (vi x - 2 > 0).
Dang thiic xay ra khi x = 3. v a y gia tri nho nhdt ciia ^(x) la 4.
booktoan.com 251
20.
a) Cdch I. TiX ddng thiic (a^ +b^ +e^)ix^ + y^ + z^) = = iax + by + czf + iay - bxf + ibz - cyf + iaz - ex)^ de dang suy ra (a^ +b'^ + c^)(x^ + y^ + z^)>iax + by + czf ay = bx
. t. a b e bz = ey tuc la — = — = — X y z az = ex
Dang thiie xay ra khi
Cdch 2 2
2 2
2 2
2 2
(ax + by + ez) = a x + b y + c z + 2abxy + 2acxz + 2bcyz < ^2_^2 ^ ^2^2 ^ ^2^2 ^ ^2^2 ^ ^ 2 ^ 2 ^ ^2^,2 ^ ^2^2 ^ ^2^2 ^ ^2^2
= (a^ +b^ +c^)(x2 + y^ + z^).
b) (x + 2y + 3z)^ ^i\.x + 42.42y +
S.Szf
< (x^ + 2y^ + 32^)(1 + 2 + 3) = 6.6 - 36. Vi vay |x + 2y + 3z| < 6. 'x + 1 < 2x - 3 21.
a) I
4
> x +3
x>4 X<-1.
He vo nghiem. b)0
X . .^.\i Taco (I)
+ rnx>0' x-2<0
<=>-^
fmx>-l x<2
(1) (2)
Gpi tap nghiem eua (1) va (2) Idn lupt la 5i va 52- Khi dd ^2 = (-f» ; 2]. - Ne'u m = 0 thi 5] = 0 nen he (I) vo nghiem '.S = 0.
booktoan.com
252
— ; + 00 va m )
- Neu m > 0 thi 5,
< 2, nen tap nghiem ciia he (I) la m
S= - - ; 2 | . m - Neu m < 0 thi 5, ^
-co ; l^
1 ta can phai so sanh m)
m
1 1 ( + Neu m < - - thi < 2, nen 5 = -oo ; 2 m [^
23.
vdi 2.
\] m
> 2, nen 5 =(-oo ; 2]. + Neu m > - - thi 2 m a) Tap nghiem S = ^1 ^ - ; 1 u ( 2 ; + co). yZ. J
^j X + 1 ^ X - 1 ^ 2x + 1 3x + 1 ^
X +5X + 2 >o^(^-^iXx-x2)^Q^ (2x + l)(3x + 1) (2x + l)(3x + 1) -5-V17
.
VOl X J =
X
(X + l)(3x + 1) - (X - l)(2x + 1) ^ Q (2x + l)(3x + 1)
-5 + 7l7
v a X;
-5-7l7
-GO
-
2 0
Ta lap bang sau :
1 2
+
+
-
-
-
2x+ 1
-
-
3x+l
-
-
-
Vetrai
+
-
+
X-X|
-
X-X2
0
()
1 3
-5 + 7l7 2
0
+
+
+
+
+
+
+ ()
0
-f 00
-
+ +
vay tap nghiem cua bat phuong trinh la S=
-CO.
-5-7l7
^
1 _ -5 + 7l7 u 2 2 '
-3; + -
booktoan.com
253
24.
a) Lap bang pha da'u gia tri tuyet dd'i nhu sau : X
-2
—00
lx+ 11
-X- 1
3lx + 2l
-3x-6
ve' trai
-4x-7
-1 -X- 1
0
0
+ 00
x+ 1
3x + 6
3x + 6
2x + 5
4x + 7
• Vdi X < - 2 , bat phuong trinh da cho trd thanh -4x - 7 > x + 7<=>x'< -2,8. Do - 2 , 8 < - 2 nen trong trudng hpp nay, ba't phuong trinh ed nghiem x < -2,8. • Vdi - 2 < x < - l , t a e d 2 x + 5 > x + 7 < = > x > 2 . Ket hpp vdi dieu kien dang xet thi khdng cd gia tri x nao thoa man. • Vdi X > - 1 ta cd 4x + 7 > X + 7 <=> X > 0. Do - 1 < 0 nen trong trudng hpp nay, nghiem cua ba't phuong trinh la x > 0. Vay tap nghiem S = (-oo ; -2,8) u (0 ; +oo). b)
-5 10 < <» x-1 x+2
1 x+2
2 Y
<
x-1
<=>
3(x + l)(x + 5) ^ _
— ^ U.
( x - l ) 2 ( x + 2)2
Lap bang xet da'u ta tim dupe tap nghifm la 5 = (-00 ; -5] u [-1 ; 1) w (1 ; +oo). 25. >• .
A\
X v=2
B 2 s C /
O
A
2\
D / . --2
Hinh 3
254
.V x\
booktoan.com
Hinh 4
a) Tap nghiem cua ba't phuong trinh ia mien tii giac ABCD (kd ca bien) (h.3). b) He vd nghiem (h.4). 26. a) Ta cd A = (m - 1) ^ - 12(m^ + 1) = - l l m ^ - 2m - 11 = -(Um^ + 2m + 11) va a - m + 1 > 0. Tam thdc ludn duong khi va chi khi A = -(llm^ + 2m + 11) < 0 <^ 1 Im^ + 2m + 11 > 0 <^ m G M. b) Ne'u m= yl2 de tha'y bieu thdc luOn duong vdi mpi x. Ne'u m^ yl2 thi bidu thiic la tam thdc c d a = y/2-m^Ova
biet thirc
A = (m-72)^-4(72-m)(2m + 3 72)-9m^ + 272m-22. f a = 72 - m > 0 Tam thiic ludn duong khi va chi khi \ (*) [A - 9m^ + 272m - 22 < 0. 2 /-ll72 Tam thiic/(m) = 9m + 2v2 m - 22 ed hai nghifm my = —-—> m2
= ^/5. 1 1 /O
Do dd/(m) < 0 khi va chi khi ^-r—
< m < 72
1 1 lo
Ke't hop vdi (*) suy ra ^-r— 27.
< m < 72 .
9yl2 a) Tam thdc ludn ludn am khi va ehi khi m > —;— ^ 4 b) Vdi m = - - , khi dd bidu thirc cd gia tri la - - > 0, do dd m = - khong thoa man. Vdi m i= - - ' khi dd bidu thiic da cho la mpt tam thiic bac hai.
booktoan.com 255
Tam thdc ludn am khi va chi khi [a = 5m + 1 < 0 |A
28.
o
m<-l.
^ (5m + 1)^ - 4(5m + l)(4m + 3) < 0
a) Bat phuong trinh tuong duong vdi
X
x(x + 4)
> 0, suy ra tap nghiem la
(-4 ; 0) u (0 ; +^). b) Bat phuong trinh dupe bie'n doi tuong duong vdi x'+l6
>0.
2 x ^ ( x - 2 ) ( x + 2)2
Suy ra tap nghiem la S ^ (2 ; +co). 29.
a)6
30.
a) Bat phuong trinh da eho ed he sd a = 3 > 0, dd ba't phUOng trinh vd ,
2
nghiem, dieu kien edn va du la : A = m - 12(m + 2) < o o m ^ - 12m-24
6 - 2 7 l 5 < m < 6 + 27l5.
b) Vdi m = 3, khi dd b^t phuong trinh trd thanh -2x - 1 > 0 va hit phuong trinh cd nghiem la x < -—• Suy ra m = 3 khdng thoa man. Vdi m^3.Di
ba't phuong trinh vd nghiem didu kien e^n va dii la : m>3
3 - m <0 A = (2m - 5)^ - (3 - m)(5 - 2m) < 0
<=>
|2m^ - 9m + 10 < 0
m>3 <=> <
2
Suy ra khdng ton tai m di bat phuong trinh da cho vd nghiem.
booktoan.com
256
31.
Dat y = x^, y > 0. Khi dd ve' trai cua phuong trinh da cho trd thanh /(y) - y ~ 2my + m - 1 . Dieu kien eiia bai toan dupe thoa man ne'u phuong trinh /(y) = 0 vd nghiem hoac ehi ed hai nghiem am. Cdch } . Do A' = 1 nen phuong trinh/(y) = 0 cd hai nghiem y^ ^ m - 1 va y2 - m + 1. Ta 'phai cd
Jm - 1 < 0 [m + 1 < 0, tire la m < - 1 . Vay phuong trinh trung phuong da cho vo nghiem khi m < - 1 . Cdch 2. Do A' = 1 nen phuong trinh/(y) = 0 cd hai nghiem phan biet. Hai nghiem dd am khi va chi khi — = 2m < 0 a - - m^ - 1 > 0, a tiic l a m < - 1 . Ta cd ke't luan nhu tren. 32.
- 3 < m < 1. Hudng ddn. Dat y = x + 1 bai toan trd thanh : Tim m sao cho phuong trinh im - l ) ( y - 1)^ - (m -5)(y - 1) + m - 1 = 0 ed hai nghiem duong phan biet, tu'c la phuong trinh (m - l)y^ - (3m - 7)y + 3m - 7 = 0 cd hai nghiem duong phan biet.
33.
a) X - ± 3. b) x € l ± 3 ; ± i
34.
a)S = 1;
U + 72
4-72^ u
;3
booktoan.com 257
Gm y. Ba't phuong trinh tuong duong vdi he : -x^ + 4x - 3 < (x - 2)^ x - 2 >0 -x^ + 4x - 3 > 0.
b) S =
-I-'
Gai y. Ba't phuong trinh tuong duong vdi :
x + l<0 35.
r (il
a ) 5 = -\hr ;
-
•
;
11
x + l>0. v •
fn
v-r
V2
V 2
Gai y. Dat t = 72x^ - 8 > 0. b)5 = (-oo ; - l ) w ( 4 ; + cx)). Gm y. Dat f = 7x^ - 3x + 5 > 0. 36.
a) Da'u hieu la tudi cac ba me d nude MT sinh con l^n dau. Don vi didu tra la cac ba me d nude Mi sinh eon lin diu. b) Tudi trung binh la 22,89. c) Bang phan bd tan sua't Khoang
Tin sua't(%)
[15 ; 19]
31,6
[20 ; 24]
35,5
[25 ; 29]
19,8
[30 ; 34]
9,6
[35 ; 39]
3,5
booktoan.com 258
d) Bidu do hinh quat (h.5)
e) Bidu dd tan suat hinh cdt (h. 6)
40 • • 3 0 •• 20 •
-rmm
10-
n 15
19 20
Hinh 5
37.
24 25
29 30
34 35
IX 39
Hinh 6
Gpi sd be nhat la a. Sd Idn nhat la a + 18 > 8. Vay cd thd xay ra hai trudng hpp sau Trudng hap J. Mau l a a ; & ; 8 ; 8 ; a + 18 (sap xe'p theo thii tu tang din). Khi dd tong eae sd lieu \a2a + b + 34= 12 x 5 = 60, suy Ta2a + b ^ 26. Vi a < 8 ; /? < 8 nen 2a + b< 24. Vay trudng hpp nay khong xay ra, Trudng hap 2. Miu l a a ; 8 ; 8 ; & ; a + 1 8 (sap xe'p theo thii tu tang din). Khi dd tong cac sd lieu la 2a + 6 + 34 - 12 x 5 = 60. Suy ra 2a + & = 26 hay ^ - 2 6 - 2 a = 2 ( 1 3 - a ) . Vay b chan, tiie la b cd dang b = 2c. Suy ra c - 13 - a. Vi 6 < a + 18 va a = 1 3 - c n e n 2c < 1 3 - c + 1 8 = 3 1 - c . Vay 3c < 31 hay c < lO.Via < 8 n e n c > 1 3 - 8 = 5. Khi dd ft > 10 > 8. Tdm lai 5 < c < 10. Nhu vay ta cd 6 miu thoa man dieu kien da neu la { 1 3 - c ; 8 ; 8 ; 2 c ; 31 - c } trong dd c G { 5 ; 6 ; 7 ; 8 ; 9 ; 10}. Cu thd la cae m i u { 8 ; 8 ; 8 ; 10; 26),
(5
8 ; 16; 23},
{ 7 ; 8 ; 8 ; 12; 2 5 | ,
{4
8 ; 18; 22},
(3
8 ; 2 0 ; 21}.
{6; 8 ; 8 ; 14; 24),
booktoan.com 259
38.
3sin(a - P) = ^iniP - a + a) = sina eos(a - p) - sin(a - P)cosa, tif do ta cd ^ sin a (3 + cosa)sin(or - p) = sina cosia - p) (*), vay tan(a - P) = ^ 7 iChu y. cos(a - p) ^ 0 vi neu cos(« - y^ = 0 thi tir (*) ta suy ra smia-
39.
p) = 0, vo li).
Ta cd : cos"x+ 2cosaeosy9cosx= cos/[cos(7c- ia + p)) + 2 c o s a c o s ^ = cos/[-cosa c o s ^ + sinorsin^+ 2cosacos;?] = c o s / c o s ( « - y ^ T
2
= -cos(a + ^ c o s ( a - > ^ = sin'asin ^ - c o s
''
2
acos'p
= sin^asin"y5-(l -sin^a)(l -sin'^y^^-l + s i n " a + s i n V = l-cos"a-eos"yft 40.
Dat t = t a n ^ , thi 4.^—^^^ 2 1 + eos a
? — + 3 = 4r^- 2(1 + t^) + 3 = 2r^ + 1, -> a cos' — nen gia tri nho nha't dat dupe la 1 khi t = 0.
41.
a) a ^ ~ + kn \ a ^ — + k— : a ^ — + k— voik
&I,.
Hudng ddn. Cd the vie't miu thanh (cosa + cos7a) + (cos3a + cos5a) = 2cos4a(eos3a + cosa) = 4 cos a cos 2 a cos 4a. b) Hudng ddn. Viet tu thiic thanh 2sin4a(cos3a + cosa). 42.
Diing cdng thiic ha bac va cong thdc bien ddi tong thanh tich.
booktoan.com 260
MUCLUC Trang Chi/ang I. MENH DE - TAP HCJP
5
A. NHUTNG KIEN T H Q C CAN NHCJ
5
B. DE BAI
6
§1. Menh de va menh de chira bien
6
§2. Ap dung menh de vao suy luan toan hpc
9
§3..Tap hop va cac phep toan tren tap hpp
11
§4. Sd gan dung va sai sd
12
Bai tap on tap chUdng I
12
Gidi thieu mpt sd cau hoi tr&c nghiem khach quan
14
C. DAP SO - H I J 6 N G D A N
LCfl GIAI
ChLfOngll. HAIVISO
16 26
A. NHLfNG KIEN T H Q C CAN NHd
26
B. DI BAI
29
§1. Oai cUOng ve ham sd
29
§2. Ham sd bac nhat
32
§3. Ham sd bac hai
34
Bai tap on tap chucfng II
36
Gidi thieu mot sd cau hoi trac nghiem khach quan
37
C. O A P S O - HLTCJNG D Abooktoan.com N - LC(I GIAI
39
261
Chuong III. PHUONG TRINH BAC NHAT VA BAC HAI A. NHIJNG KIEN
THCTC
CAN NHCJ
55 55
B. DE BAI
58
§1. Dai cUOng ve phUdng trinh
58
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59
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62
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63
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66
Bai tap on tap chUdng III
67
Gidi thieu mpt sd cau hoi trac nghiem khach quan
70
C. D A P S O - HaCJNG D A N
71
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Chaang IV. BAT DANG THlTC VA BAT PHUONG TRINH A. N H Q N G K I E N T H Q C CAN
NHCJ
99
B. D E B A I
102
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106
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107
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109
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110
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111
§7. Bat phuong trinh bac hai
112
§8. Mpt sd phuong trinh va bat phuong trinh quy ve bac hai
114
Bai tap on tap chuong IV
116
Gidi thieu mpt so cau hoi trac nghiem khach quan
120
C. DAP SO - HUdNG D A N
122
LCfl GIAI
booktoan.com 262
99
Chuang V. THONG KE
1^2
A. N H Q N G K I E N T H Q C C A N NHCJ
172
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173
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173
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175
Bai tap on tap chuong V
180
Gidi thieu mpt sd cau hoi trac nghiem khach quan
181
C. D A P S O
HLTdNG D A N - LCJI GIAI
Chirang VI. GOC LUONG GIAC VA CONG THLfC LUONG GIAC
183 193
A. N H Q N G K I E N THCTC C A N NHCJ
193
B. DEBAI
195
§1. Goc va cung lupng giac
195
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195
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200
§4. Mpt sd cong thtfc lupng giac
204
Bai tap on tap chuong VI
206
Gidi thieu mpt sd cau hoi trac nghiem khach quan
208
C. D A P S O - HUCJNG D A N - LC!l GIAI
210
Bai tap on tap cuol nam
238
A. DE BAI
238
B. D A P S O - HLTCJNG D A N
L0\ GIAI
243
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